The Seagulls performed admirably, struck the woodwork in the first half and deserved their equaliser at the start of the second half.
They then defended resolutely as we started to dictate possession and kept clear-cut chances to a minimum - it took a spectacular winner from Keano eight minutes from time to clinch the win.
Jermain Defoe was back in the starting line-up as Martin Jol made changes from the draw at Old Trafford with Stephen Kelly, Anthony Gardner and Michael Brown in for Noureddine Naybet, Noe Pamarot and Rohan Ricketts.
Brighton struck the first notable blow of the game when Richard Carpenter's 20-yarder cracked the crossbar after 15 minutes.
Kelly and Robbie Keane tried their luck, Carpenter shot wide and we then hit the woodwork ourselves after 31 minutes, Erik Edman picked out Kelly, he nodded the ball into Keane who dinked it over Kuipers but onto the crossbar.
Daniel Harding lashed wide as the visitors continued to probe and Gary Hart might have done better after chasing onto Knight's flick but chose to try his luck rather than square and lashed into the crowd.
The goal arrived soon after. It was almost slow motion as Ledley King flicked Reto Ziegler's cross towards goal and, with Kuipers stranded, it floated home at the unguarded far post.
Brighton hit back though - Kelly was forced into a fine tackle to deny Knight before the nippy striker looped a header that Robinson had to tip over.
The first half would have given Mark McGhee's men belief and their performance was rewarded just two minutes into the second half. Reto Ziegler was somewhat harshly punished for a foul outside the box and sharp-shooting Carpenter this time got it spot on, curling wide of the wall and beyond Robinson into the far post.
We started to push on more and more and Kuipers did well to cling onto Defoe's effort but Brighton might have snatched the lead when Hart flicked Carpenter's free-kick towards goal and Robinson was forced into a superb reflex save to his left.
It was all us now but we couldn't quite carve out an opening and when Ziegler tried his luck from distance only for Kuipers to fingertip his effort from the top corner it looked like one of those days.
But Keane wasn't having it and, receiving a pass from Edman, held off Butters with his back to goal before spinning and volleying into the top corner from 18 yards. A goal worthy of winning any cup tie.
Mark Yeates, on for his home debut, left Brighton's defence mesmerised a couple of times in the final minutes as we finished on top, but credit has to go to the Seagulls for what they contributed to the game.
GUIDE TO THE ACTION
4mins: Kelly fed Defoe, back to goal, spun and shot but Butters there to block.
7mins: Hart shot from 22 yards looked like it was going wide but Robinson down low to save anyway.
15mins: WOODWORK. Sweet strike by Carpenter from 22 yards hit top of the crossbar.
20mins: Long ball found Kelly at far post, couldn't quite get it under control and forced out for corner.
21mins: Keane dribbled into box, turned inside Hinshelwood, fired high and wide.
26mins: Carpenter volleyed loose ball wide from 25 yards.
29mins: Lovely pass from King released Keane, good defending from Virgo to ease him out as Kuipers came to collect.
31mins: WOODWORK. Short free-kick from King to Edman, floated ball in to Kelly who headed into Keane's path, dinked over Kuipers but hit crossbar and bounced to safety.
33mins: Harding fired wide from 30 yards.
35mins: Knight won flick against Gardner to set Hart through, wide right, Edman tracking, should have squared but lashed wide.
39mins: GOAL. Short corner from Brown to Ziegler, King met cross and looped header beyond Kuipers into far corner.
Spurs 1, Brighton 0
44mins: Ball over the top sent Knight away, left side, Kelly's pace got him back and produced superb block tackle.
Injury time: SAVE. Looping header from Knight forced Robinson to tip over just under crossbar.
Half-time: Spurs 1, Brighton 0
47mins: GOAL. Carpenter curled 25-yard free-kick around the wall and beyond diving Robinson.
Spurs 1, Brighton 1
51mins: YELLOW. Carpenter for stopping quick free-kick.
52mins: Defoe curled free-kick over from 22 yards.
54mins: Good passing move across the pitch and down the right, Mendes' cross controlled by Brown, shot blocked.
55mins: SAVE. Defoe cut in from left flank, low shot taken well to his left by Kuipers.
63mins: YELLOW. Edman for kicking ball away.
64mins: SAVE. Harding drove in free-kick, Hart diagonal run and flicked header, Robinson superb reflex save to tip wide.
70mins: Another decent move down the right but cross cleared by Brighton.
72mins: SUB. Marney for Kelly.
74mins: YELLOW. Hart for foul on Brown.
74mins: Gardner tried his luck from 30 yards, flew wide.
77mins: SAVE. Ziegler shot arrowing into top left corner until Kuipers fingertips flicked it wide. Mendes then fired wide.
78mins: SUB. Yeates for Mendes.
82mins: GOAL. Edman dinked ball into Keane who held off Butters and hit overhead volley into top corner from 18 yards.
Spurs 2, Brighton 1
88mins: Yeates onto Keane's pass, twisted and turned in box, Hinshelwood back to block just as it seemed to open up.
Injury time: Yeates twisting and turning again, eventually drew three defenders and shot blocked.
Full-time: Spurs 2, Brighton 1
Spurs (4-4-2): Robinson; Kelly (Marney, 72), Edman, Gardner, King; Mendes (Yeates. 78), Brown, Carrick, Ziegler; Keane, Defoe. Subs: Fulop, Pamarot, Ifil.
Brighton: Kuipers, Mayo, Hart, Oatway, Carpenter, Butters, Reid, Virgo, Hinshelwood, Harding, Knight. Subs: May, Watson, Nicolas, Jones, Hammond.
A moment of FA Cup magic from Robbie Keane - the 100th club goal of his career -ended Brighton's resistance and sent us into the fourth round on Saturday.