The north London club were cruising thanks to two goals from Tomas Rosicky until Per Mertesacker conceded a penalty and was sent off for a foul on Shane Long, with James Morrison netting the resulting spot-kick.
Arsenal had to dig in to withstand a late onslaught from the Baggies, and Wenger was delighted with the strength and resolve displayed by his side after they were reduced to ten men.
"It was a difficult game against a good West Brom side who were really up for it," Wenger explained after the match.
"They did fight until the last second and we have shown two aspects of our game today. One was very in control and with technical quality and one that is less known with fighting spirit, resolute attitude and battling qualities.
|WENGER BELIEVES IN TOP FOUR FINISH
|4/9||Arsenal are 4/9 with William Hill to finish in the top four
"I believe since November we have been on quite a good run in the championship. We made 22 points out of the last 27 but we know that our target is to do that until the end of the season."
Wenger reserved special praise for match-winner Rosicky and a superb cameo performance from the Czech Republic international.
"He was fantastic," the French boss continued. "He's an important player for us. Unfortunately he was out injured for a long time but every year he gives us a little push at the end of the season which is, of course, very important.
"I hope he finishes his career with us, he still has one year on his contract. I rate him very highly."
Baggies manager Steve Clarke, meanwhile, took little satisfaction from the result even though his side gave a good performance and were unlucky not to grab a late equaliser.
"I thought we played well today," said the Scottish boss after the match. "I thought first half we were good as well. They scored with one of the very few chances that they created.
"The start of the second half was crucial for us to get back in it. We gave away another soft goal - a bad goal to lose - and at 2-0, it's difficult to chase the game.
"We made a couple of changes, we had a go and the penalty and obviously the red card changes it and gives us a chance to get something from the game."