Netherlands international Van Persie volleyed home magnificently in the third minute of stoppage time to claim his second of the match and take his season's tally to 31 in all competitions.
Van Persie's last-gasp winner saw Arsene Wenger's side cement their grip on fourth place, and moved the Londoners to within four points of third-placed Tottenham Hotspur, who play Manchester United tomorrow.
But the defeat has left Liverpool with a mountain to climb if they are to secure a return to Europe's elite next season, with ten points now separating them from Arsenal in fourth. The defeat was cruel on the Merseysiders, who had dominated throughout only to be denied by a string of superb saves from Wojciech Szczesny and the woodwork.
Van Persie admitted Liverpool had been unlucky to lose. "I don't think we deserved it if I'm being honest because Liverpool played better but to nick it like we did, we can only be pleased," the striker said.
"We are ten points ahead of Liverpool now and Spurs play tomorrow against Man Utd. We will have to wait and see. It is a big win, a massive win."
Polish international Szczesny had needed to be alert from the outset, reacting swiftly in the tenth minute to snuff out a break from Luis Suarez before dealing with a follow-up from Stewart Downing.
Szczesny then pulled off a superb double save as Liverpool missed their sixth penalty of the campaign after the Arsenal goalkeeper was adjudged to have brought down Suarez following a neat one-two with Dirk Kuyt.
Van Persie claims a double
Kuyt's spot-kick was firm but Szczesny guessed correctly, diving to his right to block before springing back across goal to deny Kuyt's attempted rebound.
It was only a temporary reprieve, however, as Liverpool took the lead on 23 minutes, Koscielny turning Jordan Henderson's whipped low cross into his own net past Szczesny. Two minutes later and Liverpool were threatening again, Suarez's shot denied by the foot of the post.
But Arsenal were back on level terms in the 31st minute with their first genuine effort of the game. Bacary Sagna sent in a superb deep cross from wide on the right and Van Persie held off the attentions of Jamie Carragher to head past Pepe Reina.
Liverpool struggled to carve out as many clear chances after the break but the Reds did squander a golden chance to go 2-1 ahead on 70 minutes. Kuyt crossed from the right to find an unmarked Martin Kelly only for the defender to miskick from point-blank range with the goalmouth gaping.
A draw seemed the likeliest result but deep into injury time, provoked by a second-half injury to Mikel Arteta, Alex Song chipped into the Liverpool area and Van Persie volleyed home on the full to clinch a dramatic win.