A Gareth McAuley own goal gave the Reds the perfect start on nine minutes, but the hosts had to wait until the final minute for a second strike from substitute Robin van Persie to seal the win.
Sir Alex pointed to the heaviness of the pitch as one of the factors behind United's somewhat sluggish display at times, but the boss was happy to see his side record another victory and maintain the seven-point advantage over Manchester City.
"The pitch was a problem for us with the way we pass the ball - it slowed our game down a bit," the boss told MUTV.
"We did well in the first half, we dominated at times, but you must give credit to West Brom who had more of the game in the second half and were a threat with their crosses. They were very resilient and stubborn and worked really hard when it came to defending.
"They had quite a bit of possession into the second period so we decided to bring Robin and Paul Scholes on just to settle us down a bit and they did that well, and of course Robin scored a fantastic goal. It's a good result considering how heavy the pitch was - that's two games in four days on it and it's not easy. The slowness of it killed our speed really."
Sir Alex was pleased to see Shinji Kagawa back in action after two months out injured and he was full of praise for fellow summer signing van Persie who once again stole the headlines with a stunning late strike.