The in-form Dutchman took his tally to 19 for the season with a superb left-foot strike in the 70th minute which proved enough for all three points and took Arsenal back into the top four of the Premier League.
Everton had defended well and looked dangerous on the break, but were undone by a moment of pure genius, which was applauded by Arsenal's record goalscorer Thierry Henry, watching on from the stands.
Gunners legends from Frank McLintock, double-winning captain in 1971, along with striker Charlie George, record appearance holder David O'Leary, 1989 title-winning manager George Graham, through to Invincibles Robert Pires, Jens Lehmann and Henry - who yesterday witnessed a bronze statue of himself being unveiled outside the ground - all helped make up a guard of honour as Van Persie led the current side onto the field.
Once the pre-match formalities were over, Arsenal - looking for a sixth win in eight league games - started brightly, with Theo Walcott and Ivory Coast forward Gervinho looking lively down the right.
Everton, who had won two from their last three, had a decent spell and forced a couple of corners which caused Arsenal concern.
Arsenal should have been ahead in the 15th minute when Mikel Arteta, signed from the Toffees on August transfer deadline day, put Walcott clear into the right side of the Everton area. He squared across the face of goal but Gervinho got his timing wrong and allowed it to run wide, with Tony Hibbert eventually making a block.
Arsenal continued to press and Wales captain Aaron Ramsey was unfortunate to see his shot on the turn float just over after being played into the Everton penalty area by Alex Song.
Everton goalkeeper Tim Howard then got down well to beat away Gervinho's angled drive.
Walcott got to the byline again and then cut the ball back through a crowded six-yard box, but it was hacked clear.
There was a scare for Arsenal at the start of the second half when Walcott fell awkwardly on his knee, but the England man soon returned to the action.
The Gunners continued to press, but once more Everton had plenty of men back and covered the space well, Dutchman Johnny Heitinga enjoying a fine game.
Walcott skipped past Leighton Baines in the 56th minute and into the right side of the area, where his fierce angled drive was beaten away by Howard.
Thomas Vermaelen, playing at left-back because of injury to Andre Santos in the midweek Champions League defeat at Olympiacos, then had to be alert to put a deep cross from the left behind as Tim Cahill looked to ghost in at the far post.
Everton made a change in the 64th minute when striker Louis Saha was replaced by defender Sylvain Distin.
Despite the tactical switch Everton swept forwards, with captain Phil Neville chipping over from 18 yards.
Then out of nothing, Van Persie netted a stunning volley to give Arsenal the lead with 20 minutes to go.
Song chipped the ball across the Everton back line, where Van Persie had drifted behind his marker to dispatch a sublime left-foot strike in off the far post.
Everton substitute Conor McAleny flashed a drive just wide in stoppage time but Arsenal went on to mark the occasion with a victory as their resurgence continued.