The young goalkeeper is on loan at West Ham from Spurs - the loan only started last week - but came up trumps against his team-mates with a penalty save just three minutes from time.
Jamie Slabber was the player denied from the spot with Burch palming out the striker's kick to his left.
In doing so, Burch (pictured left) turned the tables after his mate beat him so convincingly from 16 yards to give us a 2-0 lead in the first half.
A draw at that point looked most unlikely as we dominated the first half and played some of our best football of the season to forge into that 2-0 lead.
But West Ham found a way back via a controversial penalty in the 70th minute and Malky Mackay's header seven minutes later.
The first half performance was as good as it's been in what is already an impressive season for Clive Allen's lads.
West Ham had a couple of half-chances, notably Cohen's blocked effort from a 'Sheringham' corner but otherwise all eyes were at the other end and Burch's goal.
With Mark Yeates at his impish best, we scored twice, hit the crossbar and carved out another couple of decent chances.
Eddie Silva Sousa had the first chance after five minutes as Dean Marney picked out his early run with a quickly-taken free-kick, Silva Sousa controlled well but fired over.
Yeates soon set up the next opportunity for Slabber, robbing Hayden Mullins and setting the striker on his way only for Slabber to hit the side-netting.
Marney was next to try his luck with a screamer after Johnnie Jackson rolled a free-kick to him that clipped the top of the crossbar from 30 yards.
A goal was always on the cards and arrived after 29 minutes. Yeates (pictured right) switched flanks to the right and the ball was played into his feet, he cut inside two defenders and curled in off the upright from 20 yards.
Cohen's chance was blocked after 31 minutes but Yeates was soon back on the ball, curling a free-kick from the left that deflected off the wall into Burch's hands.
The second goal arrived after 38 minutes and was all about Slabber's movement and Yeates' eye for a pass. Slabber played the ball into Yeates midway in West Ham's half and continued on his run, Yeates controlled and slid the perfect pass into his path and Slabber did the rest, a fine finish across Burch into the corner from 16 yards.
The second half didn't quite have the same sparkle. Silva Sousa wasn't alive when Mackay missed Phil Ifil's low cross, Yeates dragged a shot wide but the game came alive again on the award of what looked like a controversial penalty to the home side after 70 minutes.
Substitute Youssef Sofiane won possession in our right-back area and played in Gavin Williams, Marney slid in and came up with the ball only for the referee to point to the spot. Marney was booked, Sofiane slotted home the penalty.
It wasn't long before the Hammers were level. It was a goal of simplicity as well as Yeates gave away a free-kick, Williams floated it over quickly to Mackay who looped his header back over Fulop and into the far corner.
That looked like it might be that but Burch's heroics arrived three minutes from time. Yeates challenged Williams in the box, Williams appeared to fall on the ball and the referee awarded another penalty. Slabber stepped up, Burch saved and the points were shared.
GUIDE TO THE ACTION
5mins: Quick free-kick from Marney picked out Silva Sousa in yards of space in box, decent control but lashed bouncing ball high and wide.
16mins: Good hustling from Yeates forced a mistake out of Mullins, Slabber picked up loose ball, tried to beat Burch at near post from tight angle but hit the side-netting.
19mins: Williams curled free-kick over from 30 yards, no danger for Fulop.
24mins: WOODWORK. Super strike from Marney after Jackson rolled free-kick to him, 30 yards out, hit top of crossbar.
29mins: GOAL. Good move, ball played wide to Yeates who cropped up on right wing, cut inside a couple of players before curling into far corner from 20 yards.
West Ham 0, Spurs 1
31mins: CLOSE. Cohen's effort from a 'Sheringham' corner taken by Williams looked goalbound but blocked.
34mins: Yeates free-kick from 20 yards deflected off defensive wall straight into Burch's hands.
38mins: GOAL. Brilliant goal. Slabber played ball into Yeates who was surrounded by West Ham players but Slabber made the right run, Yeates timed the return pass to perfection and Slabber finished well into corner with his left foot from 16 yards.
West Ham 0, Spurs 2
42mins: Jackson header from Yeates' near post corner flicked off defender and just wide.
Half-time: West Ham 0, Spurs 2
56mins: Another good move, Dilevski released Ifil down the right, early cross aiming for Silva Sousa was missed by Mackay but Silva Sousa couldn't control loose ball.
57mins: Yeates turned McClenahan inside out, dragged shot wide under pressure.
70mins: PENALTY/YELLOW/GOAL. West Ham won back possession on our left flank, Sofiane played Williams, Marney slid in and came out with ball but referee pointed to spot anyway. Sofiane slotted penalty.
West Ham 1, Spurs 2
75mins: Jackson drilled shot just wide from 25 yards.
77mins: GOAL. Williams took quick free-kick after Yeates, Mackay looped header over Fulop and into far corner.
West Ham 2, Spurs 2
87mins: PENALTY/MISS. Williams penalised for handball after falling on ball in challenge with Yeates, Slabber took spot kick only for Burch to save to his left.
Line-ups: West Ham: Burch, Mullins, Brevett, Ward, Mackay (Blewitt, 80), Noble, Williams, Cohen, Ashikodi (Sofiane, 68), Reo-Coker (McClenahan, 45), Reid. Subs: Reed, Wright.
Spurs: Fulop, Ifil, McKie, Marney, Richards, Davenport, Dilevski (Malcolm, 75), Jackson, Slabber, Silva Sousa (O'Hara, 57), Yeates. Subs: O'Donoghue, Eyre, Defendi.
Robert Burch was West Ham's hero as our reserves left Upton Park wondering how they failed to bag three points on Tuesday night.