The Hammers boss is widely expected to lose his job, regardless of the result, with Martin O'Neill ready to step in to try to retain the club's Premier League status.
Arsenal, who moved to within two points of top spot, showed the Israeli's team no mercy as goals from Robin van Persie and Theo Walcott put them in command at half-time, before the Dutchman wrapped things up with a penalty on 77 minutes.
Despite guiding the east London club into the semi-finals of the Carling Cup, where they beat Birmingham here in midweek to move within 90 minutes of Wembley, the patience of owners David Gold and David Sullivan - who only appointed Grant during the summer - appears to have run out.
While the club were officially making no comment, Grant's public spat with vice-chairman Karren Brady over the collapse of Steve Sidwell's proposed transfer having appeared to have brought matters to a head.
Arsenal, meanwhile, finally produced a complete passing display to bounce back from the disappointment of their midweek cup defeat at Ipswich as they remain very much in the hunt for the title.
All cameras were trained on Grant as, after a quick exchange of words and handshake with Gunners boss Arsene Wenger, he took to the home dugout for what could well be the last time.
There was some sense of irony that Grant's likely departure came at a time when West Ham were actually showing some form, having lost just twice in their previous seven games.
Nevertheless, the Irons remain bottom and they fell behind on 13 minutes when Samir Nasri, back in the starting XI, produced a clever step over from Walcott's cross - which van Persie despatched into the right corner from 12 yards.
Irons full-back Julien Faubert was then lucky to get away with only yellow card from referee Andre Marriner for jumping in two-footed on Fabregas out on the far touchline, which left Wenger less than impressed.
Walcott shot tamely at goalkeeper Robert Green before West Ham replaced Mark Noble, who had just come back from a spell on the sidelines after a hip problem, with former Gunner Luis Boa Morte.
There was a let-off for Arsenal when Johan Djourou's backpass was too short and Carlton Cole darted in on goal - but stand-in keeper Wojciech Szczesny made a fine block and Hines smashed the rebound over from a tight angle.
Van Persie was within inches of making it 2-0 when Arsenal broke quickly through Nasri and his first-time pass into the left of the penalty area was crashed against the base of the upright by the Dutchman.
West Ham were in danger of being overrun and they doubled their lead five minutes before the break.
Van Persie latched onto Fabregas' ball down the left side of the box, which the Dutchman pulled back from the goalline to the centre, where Walcott crashed it into the roof of the net.
Chants of 'You're getting sacked in a hour' rang out from the away end as Grant stood at the edge of his technical area, hands in pockets.
It was more one-way traffic at the start of the second half, van Persie testing Green at his near post and then firing over from 20 yards.
However, Szczesny had to be alert to tip away Freddie Sears' angled drive, which looked set for the top corner after a quick break.
Wayne Bridge, making his debut on loan from Manchester City, tripped Walcott as the England winger nipped into the penalty area. Marriner pointed to the spot on 77 minutes and Van Persie slotted the ball into the bottom left corner.
Irons fans had seen enough, hundreds leaving for the exits, with Grant probably soon to follow.
At the final whistle the Israeli shook hands with his players as they left the pitch, before applauding the fans and throwing his lucky claret and blue scarf into the stands in what appeared to be a farewell gesture.