Fulham debutant Kagisho Dikgacoi received his marching orders in the 42nd minute for an apparent head-butt on Scott Parker after Carlton Cole handed West Ham the lead.
The sending off gave Fulham a difficult task in the second half but they were level almost immediately after the restart as Mathew Upson gifted the away side a penalty.
The centre-half clumsily brought down Diomansy Kamara inside the area in the first minute of the half and Danny Murphy expertly dispatched from the spot to even up the score.
Just 10 minutes later Fulham were in front, to the disbelief and dismay of the home fans, as Zoltan Gera volleyed in at the back post on 57 minutes.
The home side were reduced to long-range efforts thereafter as Fulham defended well despite being a man down.
Their persistence finally paid off in stoppage time as substitute Stanislas saw his deflected effort from outside the box fly past Mark Schwarzer in the Fulham net to grab a point for the hosts.
The result leaves the Hammers in the relegation zone and will do little to lift the pressure that is building on manager Gianfranco Zola.
ChangesUpson returned to the heart of West Ham's defence after missing the 3-1 defeat at Manchester City with a calf injury while Zavon Hines came in for Radoslav Kovac.
West Ham dominated the first half and Fulham were somewhat fortunate not to be more than 1-0 down as they struggled to cope with Cole up front and Alessandro Diamanti's deliveries from the right wing.
After Brede Hangeland had been booked for a clumsy challenge on Julien Faubert, it was Diamanti's vicious free-kick that allowed Cole to bury his header past Schwarzer at the near post.
Diamanti almost carved out a carbon copy minutes later when he curled a corner to James Tomkins at the far post but the unmarked West Ham defender headed wide.
Murphy's quick free-kick released Andrew Johnson into the West Ham box but Robert Green raced off his line to close the angle and the Fulham striker pulled his shot across goal and wide.
But it was a rare opening for Fulham, who were on the back foot.
Cole almost picked out Hines after an incisive passing move and he then sent Mark Noble through on goal with a deft touch from Luis Jiminez's lofted pass.
Parker changed the point of attack with a driving run into the box and then saw a 30-yard drive saved after Fulham had again failed to deal properly with another Diamanti cross from the right.
Fulham were then reduced to 10 men when Dikgacoi was sent off for an off-the-ball clash with Parker.
West Ham were playing slick, one-touch football, which allowed them to make use of the extra man to create two more chances before the interval.
Noble's shot from inside the box was tame before Herita Ilunga slipped Hines into space and the young winger unleashed a sweet right-foot drive that flew just wide of the post.
Second periodWest Ham were playing with such attacking verve it seemed victory was a formality but Fulham emerged in the second-half to produce the perfect example of how to play a man down.
Chris Baird came on for Johnson to shore up the midfield and within 30 seconds they were level as Diomansy Kamara bustled into the area and was brought down by Upson.
Murphy beat Green from the penalty spot and West Ham began to lose their composure.
Green reacted smartly to block Paul Konchesky's low free-kick but, from the resulting corner, Gera emerged unmarked at the far post to put Fulham head with a cool, side-footed finish.
Julien Faubert was booked for throwing the ball away after conceding a soft free-kick and the frustration began to grow around Upton Park.
Zola made two changes just after the hour mark, with Parker replaced by Valon Behrami and Stanislas introduced for Hines.
Behrami brought urgency to West Ham but Diamanti resorted to shooting on sight and Stanislas opted for a speculative drive from distance that flew over the bar.
West Ham finally found the breakthrough two minutes into added time when Stanislas worked space inside the Fulham box and saw his shot deflected off Aaron Hughes past Schwarzer.
West Ham UnitedTeam StatisticsFulham2Goals211st Half Goals07Shots on Target612Shots off Target410Blocked Shots18Corners413Fouls151Offsides13Yellow Cards20Red Cards186.8Passing Success7520Tackles1575Tackles Success73.364.4Possession35.654.7Territorial Advantage45.3