Manuel Almunia started in goal for the Gunners, with Eduardo Da Silva leading the attack as Arsene Wenger's side looked to protect a 2-0 lead from the first leg in the Czech Republic.
Just seven minutes in the Gunners took the lead on the night and extended their aggregate advantage as Theo Walcott fired a cross in from the right and Tomas Rosicky arrived right on cue to bury a right-footer from 12 yards.
Robin Van Persie almost added a second seven minutes later as he reached Eduardo's deep cross, but could only knock his effort over the top from close range.
Sparta finally threatened 25 minutes in as Marek Kulic's shot was deflected narrowly over the top of Almunia's goal.
Eleven minutes before the interval Jan Rezek should have levelled the score on the night when played clean through on the right, but he lifted his shot over Almunia and wide of the post.
After the restart Sparta failed to make any more of a fight of the tie, showing little in the way of attacking intent.
Arsenal also seemed content to let the time tick away without incident, with a three-goal cushion more than enough to ensure that they would join Manchester United, Chelsea and Liverpool in the group stage.
The Gunners put a gloss on the scoreline late on in the game as they added two more goals to complete a resounding aggregate win.
Firstly Eduardo skipped away from his marker down the left and teed the ball up for Cesc Fabregas to rifle home a low, right-footer from 15 yards.
Eduardo then got himself on the scoresheet right at the death as he touched home Denilson's ball in from the right to complete a satisfactory evening for Wenger's men.