Rosicky broke the deadlock at the Hawthorns with his first league goal of the campaign and then added a second early in the second period.
West Brom midfielder James Morrison reduced the arrears from the penalty spot after Per Mertesacker was red carded for bringing down Shane Long but the Gunners held on to claim the three points.
Victory enabled Arsene Wenger's side to climb above Chelsea into fourth spot and they are now only a point behind Spurs with their two main rivals both in action on Sunday.
Arsenal needed a goal-line clearance from Rosicky to prevent Albion taking the lead after 12 minutes. Chris Brunt's corner picked out Claudio Yacob's run to the near post and his glancing header beat Lukasz Fabianski only for Rosicky to head away.
After 20 minutes Arsenal broke the deadlock through Rosicky's first Premier League goal of the campaign. Mikel Arteta's long pass was seized upon by Gervinho who turned past Gareth McAuley before crossing for Rosicky who headed home at close range.
Aaron Ramsey should have doubled Arsenal's lead two minutes before half-time but he poked the ball wide of the far post after good play by Giroud and Santi Cazorla.
After 51 minutes Rosicky doubled Arsenal's lead with his second goal of the game. Ramsey's pass was volleyed by Rosicky with plenty of power and Foster was unable to hold the shot. Rosicky was the first to react to the rebound and made no mistake from eight yards out.
Baggies boss Steve Clarke responded by bringing on strikers Lukaku and Markus Rosenberg in place of Graham Dorrans and Jerome Thomas respectively.
Albion were given a lifeline after 71 minutes when last defender Mertesacker was sent off for bringing down Long in the box and the resulting spot-kick was converted by Morrison. The closest Albion came to levelling was when McAuley headed just wide from a Brunt corner. Lukaku curled a shot just wide and Long fired over the bar but Arsenal held on.
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