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Van Persie showed superb skill and speed to fasten on to a pass from Ashley Cole before keeping a cool head and walking the ball around Brad Friedel before tapping into an empty net.
The goal came just seconds after Blackburn had been forced into a defensive reshuffle after they lost defender Dominic Matteo through injury.
Blackburn's defensive plan, which has worked so well in recent weeks, was disrupted still further when skipper Andy Todd failed to appear after half-time through illness.
Rovers might well have snatched a point though with substitute Brett Emerton and midfielder Steven Reid both going close in the final minutes.
Blackburn had the better of the early play and David Thompson should have done better when he skipped past two defenders before dragging a weak shot wide at the post when a cross looked a better option.
Arsenal looked lively and dangerous on the break and Van Persie twice looked on as long-range efforts were deflected wide.
Blackburn's Norwegian winger Morten Gamst Pedersen produced a corner for which he is becoming renowned on the half-hour but Jens Lehmann was not caught out as he managed to tip the ball over the bar.
Lehmann then held on to a Pedersen header right on his goal-line as Blackburn began to get up a head of steam backed on by their vociferous fans.
A free-kick from Reid flew over the bar but Rovers' hopes of finding a way through and snatching victory were thwarted with Van Persie's winner - a goal of stunning quality.
With Blackburn having to reorganise at the back, Arsenal threatened to increase their lead several times in the opening minutes of the second half.
Van Persie, a real live-wire for the visitors, was in on goal after more great play by Cole, but Friedel made an outstanding save, diving low to his left to push the ball away.
Jose Antonio Reyes also forced Friedel into action before being carried off on a stretcher, but the Spaniard was soon back in the action tormenting the home defence with his ability to twist and turn in and around the box.
Van Persie seemed certain to add to his goal in the 66th minute when a superb pass from Reyes put him in the clear, but, unluckily for Arsenal, the young Dutchman's shot beat Friedel only to bounce away to safety off the crossbar.
Having won five of their previous six away matches, Arsenal continued to look dangerous as Blackburn's challenge wilted in the sunshine.
However, the introduction of Australian winger Emerton put a new spark into Rovers and suddenly Arsenal were on the back foot.
Emerton it was who latched on to a pass from his fellow countryman Lucas Neill to break the Arsenal offside trap and steam towards goal.
He chose to shoot rather than cross and was unlucky to see his effort flash across the face of goal with Lehmann stranded.
But Rovers' best chance of a point came in the dying seconds when a super move carved Arsenal open and left Reid with only Lehmann to beat.
Unfortunately the Irish international panicked and his shot flashed a yard wide when he really should have at least hit the target.
MAN OF THE MATCH: Robin Van Persie (Arsenal) – Scored a goal of exquisite quality to decide a tight match. The young Dutchman also hit the bar as he helped to banish the memory of his red card at Southampton that so incensed Arsene Wenger.