Arsenal took full advantage of an uncertain performance from an injury-hit Hammers side playing in an almost funereal atmosphere at a normally boisterous ground, after a day that seemed set to be Grant's last stand.
The Gunners gave a fluent and assured display as they moved within one point of Manchester United and stayed three behind Premier League leaders Manchester City, although Arsene Wenger's side rarely needed to be at their best against such troubled opponents.
The visitors had endured a torrid week of their own, drawing at home to Leeds in the FA Cup and losing to Ipswich in the Carling Cup, but they took the lead with the first meaningful attack of the game after a lapse by Bridge, who endured a disastrous Hammers debut.
The left-back, who joined the club this week on a reportedly extravagant salary despite only starting one league match this season for Manchester City, was exposed for all three of Arsenal's goals.
Under-pressure Grant salutes fans
For the first, the 30-year-old allowed Walcott space on the right to send in a pass that was expertly dummied by Samir Nasri, allowing Van Persie to swivel and smash right-footed past Robert Green.
Walcott should have doubled the Gunners' lead after wriggling free of James Tomkins on the break but the winger hit a weak shot straight at Green.
The depleted Hammers, who were without the injured Scott Parker and lost Mark Noble early in the game, were offered mild encouragement by a nervy start from the Gunners' inexperienced backline, with a mistake from Johan Djourou almost leading to a Hammers equaliser.
The centre-back's sloppy backpass let Cole through on goal but Szczesny parried superbly and Zavon Hines blazed a follow-up chance over the bar.
It looked a costly miss because at the other end it seemed the slick Gunners could score at will.
Nasri should have done better with an effort that went straight at Green and Van Persie hammered a shot against the post from 12 yards.
But the visitors did not have to wait much longer for their second goal as Van Persie, played onside by Bridge, burst to the byline and pulled the ball back for Walcott to blast into the roof of the net from six yards.
Wenger sympathises with Grant
The Hammers had an instant chance to reply but Cole sent a header off target after being set up by an excellent Freddie Sears cross.
A relaxed Arsenal continued to control the game after the break and peppered the Hammers goal, although the hosts almost grabbed a fortuitous lifeline when Sears' miss-hit cross was tipped just wide by Szczesny.
The last flickers of a Hammers challenge, and perhaps of Grant's reign at the club, were extinguished when the hapless Bridge hacked down Walcott in the box and Van Persie stepped up to fire confidently into the bottom corner.
West Ham had the excellent Green to thank for keeping the score respectable, the keeper parrying a late Jack Wilshere effort just wide at a rapidly emptying stadium.
There were just about enough spectators remaining to give Grant a subdued send-off at the final whistle, the ever-dignified manager applauding the fans and slinging a claret and blue scarf - which just days ago he had dubbed his "lucky" garment - into the crowd.
Arsenal manager Arsene Wenger: "We had a good, mature performance today. It became pretty comprehensive.
"I said many times that the Premier League will be exciting and interesting until the end.
"It is in our hands, we play all of the big teams at home and we have a strong run at home now."
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