Arsene Wenger made just one change to the team which lost at home to Hull City at the weekend, with Samir Nasri fit again to replace Emmanuel Eboue in midfield.
The Gunners made a bright start with van Persie extending Porto keeper Helton with a low, left-foot drive from 25 yards out 12 minutes in.
However, the visitors sprang forward within a minute, with Cristian Rodriguez meeting a right-wing cross from Tomas Costa and seeing his downward header go over after clipping the top of the crossbar.
Shortly before the half-hour mark Lisandro Lopez found space 25 yards out and rifled in an effort which was turned behind for a corner by Manuel Almunia.
From the resulting set-piece, Lopez pounced from close range but saw his prod towards goal cleared off the line by the covering Gael Clichy.
With 31 minutes played the Gunners broke the deadlock as Cesc Fabregas found Adebayor on the right and he laid the ball back to the near post for van Persie to turn it home.
Five minutes before the interval it was 2-0 as van Persie's corner from the right was met by Adebayor and his header bounced down and into the net off the underside of the crossbar.
Just three minutes into the second half it was game over as van Persie drifted away from Bruno Alves' weak challenge and poked a left-foot shot beyond Helton and just inside the far post.
Five minutes later Theo Walcott wasted a chance to add a fourth as he stabbed the ball wide from close range after Kolo Toure had played the ball back into the danger zone.
Walcott turned provider soon after as he raced clear down the right and played the ball in for Nasri, but he screwed the ball wide of target from close range.
Twenty minutes from time Arsenal were awarded a penalty for Freddy Guarin's challenge on Nicklas Bendtner and Adebayor made no mistake from the spot, sending Helton the wrong way.
With just three minutes left to play Bendtner set up Adebayor for a hat-trick, but the Togo striker somehow fired his shot just wide of the far post when clean through.
There was still time for Eboue to hit the base of a post from Carlos Vela's cutback, but the Gunners had done more than enough to secure a convincing victory.