It was our own version of the Great Escape at Christmas - the home side could have scored four times before we showed how it should be done in front of goal to secure a fifth away win already this season.
The Canaries hit the woodwork twice and had two further efforts cleared off the line as they created all the chances in the first hour, although it has to be said we were the better side in the final quarter with Michael Carrick spraying the ball to every corner of Carrow Road.
Keane pounced on some poor defending after 72 minutes before Brown lashed home from 25 yards four minutes later to all-but end it as a contest.
Even the most ardent of fans in another packed away end would have to concede that the home side deserved the lead at the break.
They hit the inside of the post and had two efforts cleared off the line, Leon McKenzie also missed a header from close range and Pedro Mendes sliced inches over his own bar.
Like the draw at the Lane in September, Darren Huckerby was the cause of most of our problems, linking up to good effect with David Bentley.
It was Bentley who struck the post after nine minutes, a shot from 20 yards from Huckerby's pass and seconds later, Mendes was left wiping his brow as his clearance from Huckerby's drilled cross flew inches over.
Mattias Jonson went close from distance and Paul Robinson had to palm out a Bentley drive before our first decent attack, Fredi Kanoute shooting across Robert Green but wide from 22 yards.
That was after 16 minutes and two minutes later Mendes was forced to clear ex-Spur Gary Doherty's close-range header off the line.
Bentley flashed a half-volley over and McKenzie might have done better with his header after Bentley and Huckerby combined again.
Our best spell came after the half-hour mark with Keane cracking a volley just wide and Carrick's sweetly-struck shot straight at Green.
Carrick had another go, this time just wide, before another goal line clearance in injury time, Noureddine Naybet the saviour after McKenzie managed to get between Robinson and King and toe the ball towards the empty net.
It didn't take long for Norwich to come close again in the second half as McKenzie rose high to meet Huckerby's corner only for Robinson to tip the ball onto the crossbar.
That was the turning point. We played the better football in the final half-hour with Carrick pulling the strings and Brown everywhere.
Thimothee Atouba had a half-chance blocked, Kanoute fired over and Pedro Mendes forced Robert Green into action for the first time, a full-length dive to keep out his volley.
Reto Ziegler came on for Atouba and was soon on the end of a fine move, shooting just wide and Mendes fired high and wide from Kanoute's pass.
The vital first goal arrived with 18 minutes left. Ziegler curled in a corner, Fleming headed back towards his own goal, Drury made a hash of his clearance and Keane drilled home from eight yards.
Four minutes later and it was all over as Brown picked the ball up off Pamarot, played a one-two with Keane, controlled and rifled the ball past Green from 25 yards.
GUIDE TO THE ACTION
3mins: Poor clearance from Doherty straight to Atouba, left wing, whipped in cross that flashed inches past far post.
8mins: SUB. Drury for injured Charlton.
9mins: WOODWORK. Huckerby passed inside to Bentley, took shot early from 20 yards, came back off inside of post.
9mins: CLOSE. Huckerby short corner, got ball back from Edworthy, drilled ball across six-yard line where Mendes sliced inches over own crossbar.
13mins: Jonson decent effort from 25 yards, dipped and swerved but always over.
15mins: Huckerby and Bentley combined again, another Bentley shot from 25 yards, Robinson palmed out.
16mins: Mendes pass into Keane, he stepped over for Kanoute who skipped away from Fleming and shot back across goal, a yard wide.
18mins: OFF LINE. Huckerby free-kick met by Doherty at far post, eight yards out, headed off line by Mendes. Huckerby causing us problems again.
20mins: Pamarot met Carrick's free-kick but headed floated wide.
26mins: Jonson, just inside box, laid ball back to supporting Bentley who took half-volley first time, just over.
28mins: Bentley long pass to Huckerby, left touchline, floated in cross that McKenzie headed over from eight yards under pressure from Naybet.
32mins: Pamarot's cross missed by Doherty sliced into air by Fleming, Kanoute won it and should have squared to Keane, hesitated for moment and chance was gone.
34mins: Keane snapshot from 25 yards flew a couple of yards over.
36mins: Carrick rocket from 30 yards straight at Green.
39mins: SUB. McVeigh for injured Jonson.
42mins: YELLOW. Safri booked for foul on Mendes.
44mins: Carrick strode away from two challenges before striking low and wide across goal.
Injury time: OFF LINE. Robinson hesitant off line as King shielded ball back, McKenzie managed to get toe in first, ball rolled slowly towards goal but Naybet managed to get back and clear.
Half-time: Norwich City 0, Spurs 0
Half-time: SUB. Mulryne for Safri.
52mins: SAVE/WOODWORK. Robinson superb save to tip McKenzie's header from Huckerby's inswinging corner onto the bar.
52mins: From that corner, chance of a break with Keane and Kanoute up against Fleming but Keane over hit pass to Kanoute and Green out to save.
53mins: Lovely move with Mendes floating pass into Brown, ball eventually worked to Atouba, shot blocked by Mulryne.
55mins: Kanoute shot high and wide from 22 yards.
56mins: Deep cross from Drury found McVeigh far post, volley into ground half-blocked by King, easy save for Robinson.
62mins: SAVE. Good passing move, Carrick always involved, eventually Keane flicked ball to Mendes who took volley first time, Green acrobatic save to his right.
62mins: Pressure leading to bit of panic in home defence, Green beaten to high ball by King, cross cut out by Doherty as Keane waited for tap in.
65mins: SUB. Ziegler for Atouba.
68mins: Kanoute control, came inside from right flank, laid ball into Ziegler who struck low effort a yard wide from edge of box.
70mins: Huckerby slid inside box to McKenzie, right angle, Edman back to make fine block tackle.
70mins: Chance on counter, Robinson long kick controlled by Kanoute, rolled ball into supporting Mendes who fired high and wide.
71mins: SUB. Defoe for Kanoute.
72mins: GOAL. Ziegler corner, Fleming misdirected header back into six-yard box, Drury miskicked attempted clearance, Keane buried from eight yards.
Norwich City 0, Spurs 1.
76mins: GOAL. Brown ball from Pamarot, played inside to Keane, took return and smashed into top corner from 25 yards.
Norwich City 0, Spurs 2
82mins: McKenzie free header from Huckerby's corner but couldn't keep on target.
83mins: Defoe onto loose ball on edge of box, went for near post, Green saved with legs.
85mins: Helveg 25-yarder saved at second attempt by Robinson.
86mins: SUB. Redknapp for Brown.
Injury time: Robinson alert to palm away Fleming's head down from Helveg in front of goal from Huckerby's corner.
Injury time: Doherty headed wide from Huckerby's corner.
Full-time: Norwich City 0, Spurs 2
Line-ups: Norwich City: Green, Edworthy, Fleming, Doherty, Charlton (Drury, 8), Jonson (McVeigh, 39), Bentley, Safri (Mulryne, 45), Helveg, Huckerby, McKenzie. Subs: Lewis, Jarvis.
Spurs (4-4-2): Robinson; Pamarot, Naybet, King, Edman; Mendes, Carrick, Brown (Redknapp, 85), Atouba (Ziegler, 65); Kanoute (Defoe, 71), Keane. Subs: Fulop, Gardner.
Referee: Mike Riley.
Robbie Keane and Michael Brown struck as we made it five wins on the trot in the Barclays Premiership at Carrow Road on Boxing Day.