The Gunners came under pressure in the second half, having opened up a 2-0 lead over their hosts thanks to a Tomas Rosicky brace.
James Morrison's penalty coupled with Per Mertesacker's sending-off forced the North London outfit to retreat at The Hawthorns, but they hung on for the precious three points in the race for Champions League qualification.
"We showed two aspects of our game today," talkSPORT quotes the manager as saying. "One side was our control of the game and our technical quality, and the other - that is maybe less known - was our fighting spirit, resolute attitude and battling qualities.
"Those qualities reflect the spirit we have in the side and I'm very pleased."
Arsenal are now in fourth place in the Premier League, one point above Chelsea having played a game more.