It took until the 13th minute for the first clear-cut chance to be created and it was the hosting Baggies who carved it out. Claudio Yacob got up highest to meet Chris Brunt's lofted pass, but his goalbound header was cleared off the line by Rosicky.
© AP Photos
Czech Republic midfielder Rosicky then played a key role in the other penalty area in the 20th minute when he broke the deadlock. Gervinho got beyond Gareth McAuley and crossed for Rosicky, who converted a header from close range to put the visiting Gunners in front.
Arsenal continued to be the dominant force after that and would have taken a 2-0 lead into the break had Ben Foster not produced a diving save to deny Gervinho in the 25th minute.
Should they hold on, Arsene Wenger's team will move up to fourth in the Premier League table.