The result increased the pressure on Chelsea and Tottenham in the battle for a Champions League spot.
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 were worthy winners despite the continued absence of injured duo Jack Wilshere and Theo Walcott and have the momentum - and no European distractions unlike their main competitors - entering the final phase of the season.
Rosicky's overall contribution was impressive but there were also key performances from Santi Cazorla and skipper Mikel Arteta.
Albion only came to life after the introduction of substitute striker Romelu Lukaku and Mertesacker's red card but they are comfortably placed in mid-table.
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.
The Gunners did not look convincing at the back and Long created an opening from himself after cutting inside but he failed to hit the target.
But after 20 minutes Arsenal broke the deadlock through Rosicky's first Premier League goal of the campaign. Arteta's long pass was seized upon by Gervinho who turned past Gareth McAuley before crossing for Rosicky who headed home at close range.
West Brom tried to retaliate and Fabianski went down at the foot of a post to smother a Chris Brunt free-kick.
But Arsenal were dominating the game and Gervinho extended Foster with a low shot on the turn.
Aaron Ramsey became the first player to be yellow carded for a challenge on the recalled Jerome Thomas and then Rosicky was booked for a late tackle on James Morrison.
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 Cazorla.
The second half continued in the same vein and Foster blocked a shot from the overlapping Bacary Sagna after being teed up by Gervinho.
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.
Koscielny was booked for bringing down Long but the Gunners were in control of the game and Lukaku's weak free-kick straight at Fabianski summed up Albion's performance.
There was a rare moment of danger for the Gunners when Fabianski blocked Brunt's near-post effort.
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.
Wenger took off Gervinho and brought on Thomas Vermaelen to replace Mertesacker.
Albion came close 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.