The lethal Dutchman volleyed the winner during eight minutes of added time having headed an equalizer in the first half after Liverpool had gone ahead courtesy of an own goal from Arsenal center-half Laurent Koscielny.
Van Persie has now weighed in with 31 goals for Arsenal this season and the latest two gave his side a rather fortunate victory that moves them three points above fifth-placed Chelsea.
"It means a lot to score at Anfield because it is a massive ground," Van Persie, who had never scored at Anfield before, told Sky Sports. "We're proud of the win and I don't think we deserved it if I'm honest. Liverpool played better but to nick it at the end is pleasing."
Liverpool, who less than a week ago won the League Cup to ensure Europa League qualification, surrendered their unbeaten home league record and remain in seventh place, 10 points adrift of the top four with a game in hand.
Leaders Manchester City can open up a five-point lead later at home to Bolton Wanderers with champions Manchester United away at third-placed Tottenham Hotspur on Sunday.
Arsenal's fans serenaded Van Persie with chants of "he scores when he wants" after his latest contribution and while that may be an exaggeration, his knack of scoring important goals for the north Londoners is extraordinary.
He had hardly been in the game when he headed in Bacary Sagna's searching right-wing cross after 31 minutes, by which time a rampant Liverpool side should have scored more than one goal.
Arsenal's Polish keeper Wojciech Szczesny had kept his side in the game early on. After clipping Uruguayan striker Luis Suarez to concede a penalty he then saved Dirk Kuyt's spot kick and the scrambled across his line to beat out the Dutchman's weak follow-up effort.
Szczesny was powerless to prevent Liverpool taking the lead after 23 minutes when Jordan Henderson's cross was sliced into the net by a panicky Koscielny.
Suarez then struck a post after Henderson's shot had been saved by Szczesny before Van Persie silenced the home fans with a stooping header from Sagna's cross.
Liverpool continued to dominate and Suarez was denied a superb solo goal after the wriggling South American had slalomed through Arsenal's defense, only for Szczesny to claw away his goalbound shot. Kuyt almost put Liverpool back in front in first half stoppage time but his close-range effort hit the post.
A lengthy stoppage after Arsenal's Mikel Arteta suffered concussion interrupted the flow at the start of the second half.
Liverpool's Martin Kelly contrived to miss an open goal with 20 minutes remaining, failing to connect from inside the six-yard box while at the other end Arsenal threatened on the break with Theo Walcott's deflected shot brilliantly saved by Pepe Reina with his legs.
A draw beckoned for Liverpool only for van Persie to bag all the points for his team when he guided a textbook volley inside Reina's near post after being picked out by Alex Song's precision pass.
"We had good chances in the second half and we knew when that chance went to Van Persie we could get the three points," Arsenal manager Arsene Wenger told Sky.
"We had two men of the match today, one completely at the back and one completely at the front."
