The visitors scored a consolation through Karl Duguid, but they put on a proud performance that kept their Premier League opponents at bay for long spells of the match.
As expected Argyle set up to play on the counter-attack. The realistic aim was a draw and a chance of a second crack at the Gunners at Home Park and the initial exchanges saw matters going to plan for Paul Sturrock's side.
Jamie Mackie even forced a save out of Lukasz Fabianski after seven minutes.
Arsenal were restricted to a couple of long-range shots for the opening quarter. When Arsenal did get a sniff at goal, keeper Romain Larrieu was on hand to save from a fine Emmanuel Eboue effort on 15 minutes.
A van Persie header flashed narrowly wide of the Plymouth goal as the Gunners looked to be getting into their stride.
Arsenal had to be on their toes in defence, with the Pilgrims keen to send testing high balls in at every opportunity.
Aaron Ramsey tested Larrieu again on 29 minutes with a powerful shot that the Frenchman fisted away to safety.
Paul Gallagher ensured that Fabianski kept warm in the 33rd minute with a dipping shot from the left side of the area that the Polish keeper tipped over for a corner.
Eboue went narrowly wide with a shot from the outside the area on 38 minutes, but the Gunners could not carve out a goal against their very resolute and determined Championship opponents.
Nicklas Bendtner's acrobatic flying header two minutes later had the Pilgrims' keeper beaten, but flew just wide of the goal.
The breakthrough came almost immediately after half-time when a Bendtner cross had been palmed away. The resultant corner was delivered into the box by Samir Nasri for van Persie to send a flying header into the roof of the net.
Arsenal's second came three minutes later after van Persie had been found on the right side of the Pilgrims area by Eboue. The Dutchman's smart turn shook off his defender allowing him to send in a low cross into the box that Bendtner turned home via a deflection off David Gray.
Arsenal's celebrations were short-lived as the Pilgrims pulled a goal back 60 seconds later through Duguid, who slid in to get onto the end of a Gray cross to stab the ball into the net.
The game was flat for much of the game after the excitement of the opening minutes after the re-start.
Ramsey got the Gunners fans off their seats in the 80th minute with a shot from the right side of the area that shaved the top of the bar and Kieran Gibbs saw his low drive roll just wide.
The pressure told and van Persie grabbed his second and Arsenal's third in the 84th minute. The Dutchman ran on to a lobbed ball forward from Nasri and lifted the ball over the keeper before smashing home from four yards out.
As Plymouth fans poured out of the Emirates Stadium, their side could have pulled another goal back, had it not been for a fine point-blank Fabianski save from Steven MacLean, following Mackie's cross.
Nasri was unfortunate not to add another in injury time when his shot from the edge of the area was deflected onto the bar and the game finished 3-1.