Zamora's somewhat fortuitous strike - diverting Jon Blondel's wayward effort home - put us 2-0 up in the second half but the home side hit back within minutes and had chances to level.
In the end the lads were left holding on for the spoils in a game they might well have wrapped up by the interval.
That first half was action packed with chances created and missed, saves made, goals disallowed and one goal on the board via Milenko Acimovic.
In truth it could and perhaps should have been more.
West Ham two goals disallowed in the opening four minutes - both correct decisions - before the goal arrived after seven minutes. Marney picked out Zamora's run, he clipped back a fine cross from the byline and Acimovic was left with the easiest of tasks from six yards.
Burch saved well from Garcia, Bywater did the same at the other end to deny Zamora and the home side had another goal chalked off, this time a closer call, when Pearson slid home Garcia's sliced shot - all that in the space of five minutes and inside the opening quarter-hour.
Still the action flowed. Acimovic was sent clear by Hughes, nipped around Bywater but decided the angle was too tight and pulled the ball back to Zamora whose effort was blocked by covering Tattam.
Pearson blazed over, Mabizela headed wide - all this and still only 26 minutes on the clock!
We had two more chances to increase our lead before the break. Bywater could only watch as Mabizela's shot deflected a yard wide before another combination between the energetic Hughes and Marney ended with Zamora heading Marney's cross over from six yards.
The balance of power certainly shifted to the home side in the second half.
Noble and Byrne tested Burch and Tattum flashed over inside the first 10 minutes but when their tails were up, we grabbed a fortunate second.
Zamora wasn't complaining though - he was in the right place at the right time to divert home a shot from Blondel that was destined for the corner flag for 2-0 with 67 minutes on the clock.
To their credit, West Ham hit back within three minutes. It was a decent goal as well as Sofiane's inch-perfect cross evaded Burch at full stretch and Blewitt nodded home.
Burch was called upon after 73 minutes, beating out Sofiane's drive and after the quietest spell of the game, the young keeer darted out again to deny the same player three minutes from time.
There was still one final chance for West Ham as Pearson went through on the left side of the box but failed to hit the target with Burch out as his feet.
MINUTE BY MINUTE GUIDE
1min: DISALLOWED. Blewitt tap-in disallowed for earlier foul.
2mins: Acimovic free-kick gathered easily by Bywater.
3mins: Acimovic blasted over from tight angle.
4mins: DISALLOWED. Byrne slid into empty net but goal ruled out for offside.
7mins: GOAL. Marney fine pass inside full-back for Zamora, he dug cross out from byline for Acimovic to nod home unchallenged from six yards. 1-0.
10mins: Byrne played in Garcia, right side of box, low shot well blocked by Burch's legs.
13mins: Lovely ball inside from Hughes released Zamora into box, slightly tight angle, Bywater did well to save one-on-one.
15mins: DISALLOWED. McMahon cross found Garcia, sliced shot fell kindly for Pearson to finish off but again flagged offside.
17mins: Pearson volley from 18 yards gathered comfortably by Burch.
18mins: CHANCE. Acimovic through on-one-one, went around Bywater, tight angle and opted to turn ball back for Zamora, shot blocked by Tattam.
21mins: Pearson blazed over from 16 yards.
26mins: Mabizela headed Blondel's corner a yard wide.
36mins: Mabizela shot took wicked deflection off Lumsden and looped a yard wide with Bywater stranded.
38mins: Decent save from Burch with feet to keep out swerving 25-yarder from Noble.
42mins: More fine work from Hughes and Marney, Marney cut back from byline, Zamora headed tricky chance over from close range.
Half-time: West Ham 0-1 Spurs
46mins: Noble curler had Burch at full stretch but keeper made save look easy.
52mins: Byrne played Garcia through on goal only to be foiled by the alert Burch.
55mins: Full-back Tattum cut in from left and flashed 25 yard shot just wide of the top right corner.
61mins: Lovely flighted ball from Acimovic released Zamora, difficult chance, left-foot volley wide from 18 yards.
67mins: GOAL. Blondel cut in from left, completely sliced right-foot shot but Zamora there to divert home. 2-0.
70mins: GOAL. Sofiane fine cross out of reach of Burch and headed home by Blewitt. 2-1.
73mins: Burch stood strong to beat away Sofiane drive at far post.
87mins: Barnard did well to meet deep Hughes cross but failed to get power on effort.
88mins: Break up other end, Burch out to save at feet of Sofiane.
Line-ups: West Ham: Bywater, McClenahan, Tattam, Byrne, Lumsden, Blewitt, Sofiane, D Noble, Garcia, Pearson, McMahon. Subs: Bunce, Forde, M Noble (D Noble, 59), Laws, Wright. Booked: McMahon, Blewitt.
Spurs: Burch, Marney, McKie, Doherty, Bunjevcevic, Hughes, Mabizela, Toda (Barnard, 64), Blondel, Acimovic (Galbraith, 68), Zamora. Subs: O'Donoghue, Hirschfeld, Foster. Booked: McKie, Mabizela, Hughes.
Bobby Zamora found the target again against West Ham as our reserves secured a 2-1 win at Upton Park on Wednesday night.