Tottenham's Paul Stalteri scored a dramatic injury-time winner as the visitors came from 2-0 and 3-2 down to deepen West Ham's relegation worries.
A Mark Noble strike and a Carlos Tevez free-kick put West Ham in control before Jermain Defoe scored a penalty after Lee Bowyer fouled Aaron Lennon.
Teemu Tainio volleyed Spurs level from Lennon's audacious flick before West Ham's Bobby Zamora headed in late on.
But Dimitar Berbatov scored a free-kick before Stalteri struck on the break.
It was a dramatic and cruel end to the game for West Ham - who looked to be on the verge of their first Premiership win in 11 games when substitute Zamora headed home Tevez's free-kick in the 85th minute.
Victory would have moved bottom club West Ham above Watford and to within seven points of fourth-from-bottom Wigan.
But their failure to consolidate on their 2-0 half-time lead means they have not won in the Premiership since 17 December and are looking increasing certainties for relegation.
Curbishley responded to last week's disastrous 4-0 defeat at Charlton by making five changes, including the return to the side of January signings Matthew Upson and Lucas Neill, who skippered the team.
And the Hammers made a positive start, with Tevez constantly prodding and probing the Tottenham defence.
Noble forced an early save from Robinson, with a crisp strike, and, after 16 minutes, opened the scoring with a shot that went through the legs of Michael Dawson.
Tevez played his part in the goal, expertly guiding a Paul Konchesky cross into Noble's path with his chest.
But Upson limped off after 11 minutes and West Ham keeper Robert Green made a series of superb early saves as Tottenham showed their cutting edge.
Green twice denied Berbatov in the space of a minute and, shortly after West Ham scored, made a brilliant double save.
The keeper parried at full stretch after Lee Bowyer diverted the ball towards his own net and then saved from Berbatov with the rebound.
Tevez was fouled by Dawson for the free-kick that he converted to extend West Ham's lead.
His floated strike went in off the crossbar close to the centre of the goal - a strike that Robinson might feel he ought to have saved.
The Argentine celebrated with a spontaneous show of joy, ripping off his shirt and jumping in the crowd - and he was engulfed amid the fans for some time.
Tottenham brought on Tom Huddlestone at the break and it was his pass that set Lennon free five minutes after the restart.
Bowyer fouled Lennon and Defoe - playing against his old club - converted from the spot with a low strike.
The momentum was with Tottenham - and their equaliser was simply superb.
Lennon guided Berbatov's cross into the path of Tainio, with a sublime backheel on the volley, and the Spurs midfielder made no mistake from 10 yards.
The game lost its shape at 2-2 but Zamora's header sparked the dramatic late finish.
Berbatov struck after 89 minutes and Stalteri scored four minutes into injury time after Green had parried Defoe's shot.
West Ham: Green, Neill, Upson (Davenport 11), Ferdinand, Konchesky, Bowyer, Noble, Quashie (Blanco 80), Etherington (Zamora 83), Tevez, Harewood.
Subs Not Used: Walker, Spector.
Booked: Noble, Tevez, Bowyer, Neill, Konchesky, Quashie.
Goals: Noble 16, Tevez 41, Zamora 85.
Tottenham: Robinson, Chimbonda (Taarabt 87), Dawson, Gardner, Young-Pyo Lee, Lennon, Jenas (Stalteri 57), Tainio, Ghaly (Huddlestone 46), Defoe, Berbatov.
Subs Not Used: Cerny, Rocha.
Booked: Chimbonda.
Goals: Defoe 51 pen, Tainio 63, Berbatov 89, Stalteri 90.
Att: 34,966
Ref: M Dean (Wirral).