David Villa fired Barcelona, hailed by Arsene Wenger as the world's best team, ahead in the first half after combining with Lionel Messi.
But their superiority faded after the break when Arsenal took control, with Van Persie starting the comeback in the 79th minute before Andrey Arshavin smashed home a superb winner.
Just 24 hours after their north London rivals Tottenham shocked Serie A leaders AC Milan, Arsenal stunned the Spanish champions.
For spells it was last season's quarter-final at Emirates Stadium all over again, with Barcelona playing some breathtaking football orchestrated by Messi.
They made it look so easy for much of this last-16 first-leg clash as they slid the ball around Arsenal's defence and sent players darting into the box.
Wenger declared yesterday that the Gunners were determined to prove they were a different team to the one humbled 6-3 on aggregate last season.
The gulf in class still appeared to exist, however, and the only wonder was that Barcelona had not made more of a purple patch, with Messi guilty of missing a glorious chance.
But the tide turned after the interval when Arsenal attacked with greater conviction and they were rewarded with their first win against the Spaniards.
Barcelona brought their vast array of stellar names to north London with Messi, Villa, Andres Iniesta and Xavi Hernandez all on display.
But Arsenal looked far from intimidated by their opponents' reputation as they set off at a frantic pace, Theo Walcott initiating the onslaught at breakneck speed.
Having carved Barcelona open once early on, Walcott threaded his way across the penalty area, evading several defenders as he ran, before finding an unmarked Cesc Fabregas.
The skipper lifted the ball to Van Persie who forced a superb save on the half-volley from Victor Valdes from a tight angle.
Sensing they needed to weather the storm against an Arsenal side looking to make an early statement, Barcelona bided their time before moving through the gears in the 15th minute.
Arsenal's defence was outmanoeuvred when Iniesta sent Messi through on goal.
Messi had timed his run perfectly, leaving Laurent Koscielny and Johan Djourou stranded, but with only Champions League debutant Wojciech Szczesny to beat he steered his shot wide of the right post.
It was a massive let-off for the Gunners, who found themselves being pushed back with increasing regularity as Messi directed a shot from a good chance straight at Szczesny.
Once again Walcott sliced through Barcelona's midfield, picking out Fabregas, but the Spaniard's cross was cleared just before Van Persie could connect.
The visitors had been threatening a goal, however, and it arrived in the 27th minute through Villa.
It all looked so easy as Messi played a perfect pass through to the Spain striker who expertly sent his shot past Szczesny.
Respite was offered by a dynamic run from Jack Wilshere, but the move ended when Van Persie took one touch too many before unleashing his shot.
Messi scrambled the ball into the net as half-time approached but the flag had been raised for offside.
Arsenal made a bright start to the second half with Wilshere just failing to get enough power on his shot to trouble Valdes.
There was greater energy about Arsenal as they staged a series of attacks, although their cause was not helped when Fabregas sent a free-kick into the arms of Valdes.
The returning Samir Nasri threatened at the near post but Gerard Pique arrived to bundle his cross into touch before Van Persie could connect and moments later the Dutchman was kept out by Valdes.
Arsenal suffered a scare in the 68th minute when Eboue let Messi creep free, but the Argentinian hit the sidenetting instead of squaring to a line of team-mates queuing up to pounce.
Arsenal continued to press as the game became increasingly open and in the 79th minute their endeavour was rewarded.
Gael Clichy did the initial damage with a superbly-lofted ball down the left channel that was gathered by Van Persie.
The Holland marksman was at a tight angle but his powerful shot slipped inside Valdes at the near post.
It was a poor piece of goalkeeping from Valdes but he could do nothing about the winner that arrived with six minutes to go.
Revelling in the space now available to them, Arsenal burst free down the right through Nasri.
Arshavin, who had replaced Walcott, arrived on the edge of the penalty area to pick up the Frenchman's pass before burying the ball into the back of the net for the goal which gives Arsenal the edge ahead of their challenging trip to the Nou Camp.