Arsene Wenger's men, comfortably ahead 2-0 from the first leg in Prague two weeks ago, virtually sealed their place in Thursday's draw in Monaco for the last 32 after just seven minutes.
Substitute Cesc Fabregas made it 2-0 after 82 minutes while Eduardo da Silva scored his first goal as an Arsenal player in the closing minutes.
Arsenal took control from the start and went ahead after Walcott whipped the ball across the face of the Sparta goal and Rosicky, unmarked at the back post, lashed the ball into the net in the seventh minute.
Fabregas, who came on as a 68th-minute substitute for Abou Diaby effectively sealed the tie and Arsenal's sixth win over Sparta in six Champions League matches since 2000 when he collected a ball from Eduardo and smashed the ball home after 82 minutes. The Croatian then completed the scoring with a close-range effort just before the end.
The opening goal effectively took the sting out of the game and Arsenal were never seriously troubled after that, completing a quartet of English clubs in the group stage along with Manchester United, Chelsea and Liverpool.