Despite being second-best for most of the game the Gunners strengthened their grip on fourth place - and lessened the chances of Liverpool catching them - thanks to the prolific Holland international.
The Arsenal captain had two efforts on target and scored both as he took his record to 25 goals in as many league matches.
Reds manager Kenny Dalglish must wish he had a striker in such form after watching his side squander a number of opportunities.
A missed penalty and two shots rebounding off the post - taking the number of times they have hit the woodwork this season to 21 - only half the story as another familiar tale unfolded at Anfield.
Arsenal were lucky to go in level at half-time and so to come away with a victory will delight Arsene Wenger and infuriate Dalglish, who suffered his first home defeat of the campaign.
The Gunners had two chances in the first 45 minutes, with Theo Walcott testing Jose Reina and Van Persie scoring.
By contrast Liverpool, without injured captain Steven Gerrard, created enough openings at the other end to have been out of sight at the interval.
After the visitors' first chance, when Jose Reina denied Walcott with a good low save after former Everton midfielder Mikel Arteta's long diagonal ball over the top of left-back Jose Enrique had put the England winger through, the visitors appeared in danger of being swamped.
Stewart Downing, so impressive in last weekend's Carling Cup win at Wembley, fluffed his shot with Wojciech Szczesny caught 10 yards out of his penalty area before Luis Suarez won a penalty when he was tripped by the goalkeeper.
Unfortunately for Liverpool their travails from the spot have been indicative of their season in general and Kuyt's effort was saved by Szczesny, who also denied the Dutchman's shot from the rebound.
Dalglish's side have now missed six penalties, excluding the two also squandered in last week's Wembley shoot-out, this season.
The Arsenal goalkeeper was powerless to prevent the opening goal, however, when Laurent Koscielny, under little pressure, turned Jordan Henderson's cross into his own net.
Liverpool should have made it 2-0 moments later when Henderson's shot was parried by Szczesny to Suarez to drilled the rebound against the post.
But just when it looked as if Arsenal were on the brink of cracking they equalised with a goal of breathtaking simplicity on the half-hour.
Bacary Sagna's right-wing cross perfectly picked out Van Persie who had got in front of Jamie Carragher and the striker made no mistake with his header.
More Suarez trickery, taking him past the quartet of Alex Song, Sagna, Koscielny and Vermaelen, saw his low shot brilliantly turned behind by Szczesny before Kuyt hit the post in added time.
Early in the second half there was a lengthy stoppage when Henderson collided with Arteta, who eventually left the field heavily concussed on a stretcher to be replaced by Abou Diaby.
Thomas Vermaelen was booked for clattering into the back of Suarez as Liverpool continued to enjoy the upper hand without really threatening.
When they did create a chance in the 69th minute it fell to Martin Kelly close in at the far post but he went with his right foot instead of his left and bundled Kuyt's cross wide.
Arsenal remained a danger on the break and Reina had to be alert to brilliantly block a Walcott shot with his legs after being wrong-footed.
Downing wasted a chance to square the ball to Suarez, hitting his cross too close to Szczesny, before screwing a volley well wide.
And the Reds were made to pay for their profligacy as in the second of eight minutes of added time Van Persie snatched victory for the Gunners by expertly volleying home Song's cross from a narrow angle.