United opened the scoring in the 13th minute, when Wayne Rooney combined well with winger Ashley Young before sending a left-footed shot into the bottom right corner.
The hosts levelled through debutant Shane Long, who made the most of United keeper David De Gea's lack of conviction, before Young masterminded a winner that West Brom defender Steven Reid deflected into his own net in the 81st minute.
Sir Alex Ferguson's side struggled away from home last season, picking up just 25 points from a possible 57 on their way to a record 19th title, and started the game in a lively fashion that suggested they were keen to travel better this term.
Encouraged by the inspirational second-half showing in the 3-2 comeback win over Manchester City in last weekend's Community Shield, Ferguson backed his youngsters to deliver on a bigger platform.
"He was a threat all the time," Ferguson told Sky Sports.
Young's confidence contrasted with goalkeeper De Gea, whose shaky start to his United career continued after his handling of Manchester City's goals last week was widely questioned.
The 20-year-old Spaniard, who arrived at Old Trafford from Atletico Madrid for around 18 million pounds as a replacement for the retired Edwin van der Sar, should have comfortably saved Long's nonchalant effort in the 37th minute.
"David should have done better, I think it was just a bit of concentration there... welcome to English football," Ferguson said.
Goalkeeping was also an issue for Chelsea on Sunday, as they were held 0-0 at Stoke City thanks to some excellent saves by Stoke's Asmir Begovic.
Following a muted first half, Chelsea picked up the pace after the break with John Obi Mikel, substitute Nicolas Anelka and Salomon Kalou all testing Begovic's reflexes.
