Rosicky scored after 20 and 50 minutes to put Arsenal in control before James Morrison gave West Brom hope by converting a 71st minute penalty after Per Mertesacker had been sent off bringing down Shane Long.
Arsenal created their first opportunity after eight minutes when a Claudio Yacob tackle rebounded to Rosicky. The Czech midfielder had time to take a touch, but instead shot first time and snatched at the ball, screwing it badly wide.
West Brom soon responded, however, and the pace of Jerome Thomas against Bacary Sagna forced a couple of corners. From the second, Yacob had a header cleared off the line by Rosicky and, when the ball was played back into the area, Long could only turn his attempt over, with Lukasz Fabianski struggling.
After 20 minutes, Arsenal took the lead when a long, raking pass from Mikel Arteta held up for Gervinho to chase down. The Ivorian turned Gareth McAuley inside and out before sending a shot on goal, which was going wide, but Rosicky intelligently directed the ball home with his head from six yards.
Fabianski did well to hold a low Chris Brunt free-kick as West Brom tried to react to going behind, but it was Ben Foster who was forced into the next saves as he comfortably kept out attempts from Gervinho and Aaron Ramsey.
Ten minutes before the break and the home side had an appeal for a penalty as Laurent Koscielny planted a foot into the midriff of Yacob, but Howard Webb turn down the appeals.
Just before half-time, Ramsey missed a glorious chance to double Arsenal's lead when, after intricate work from Santi Cazorla and Olivier Giroud, he was through on Foster put somehow poked the ball wide.
Arsenal started the second half by again dominating possession, and Foster saved smartly from the adventurous Bacary Sagna before Rosicky scored his second. Cazorla played a beautiful ball into Ramsey and he fed the Czech, who tucked away the rebound after his first shot was saved.
West Brom finally brought on Romelu Lukaku with half an hour remaining, and within five minutes he had forced Fabianski into a save from a free-kick, and the Polish keeper then saved very smartly from a stinging Brunt volley.
The hosts were back in the game with 20 minutes remaining when Mertesacker was sent off for a rash challenge on Long and Morrison drilled the resulting penalty through Fabianski.
Suddenly, it was backs to the wall for Wenger's side as West Brom pressed, with Lukaku the most dangerous threat.
McAuley headed just wide from a corner and then Lukaku somehow steered the ball wide from eight yards before Long blazed over from close range as Arsenal grimly held firm.