Toure opened the scoring after nine minutes as his shot went through the legs of David Silva before Craig Dawson's deflection took the ball past Boaz Myhill.
The 32-year-old midfielder added his second 15 minutes later as he met Wilfried Bony's lay-off with a neatly curled effort into the corner of the net.
Kompany wrapped up the victory in the second half with a fine header at the near post from Silva's corner.
City's £49 million summer signing Raheem Sterling was handed his official debut, while Sergio Aguero made a late appearance from the bench.
West Brom, meanwhile, handed debuts to Rickie Lambert, James McClean and James Chester.
City began the match on the front foot and were quickly rewarded for their spell of pressure.
Having received the ball from a Jesus Navas cutback on the edge of West Brom’s box, Toure scuffed an effort that trickled past Myhill following deflections from Silva and Dawson.
Toure’s second was the result of a well-worked move by City as the Ivorian received Bony’s perfectly weighted lay-off on the edge of the area and placed his effort beyond the reach of Myhill.
Another Dawson deflection saw Bony’s shot drift wide shortly after the half-hour mark, while Sterling missed a golden opportunity to mark his debut with a goal as he failed to beat Myhill after being sent through one on one.
West Brom, who announced the club-record signing of Salomon Rondon from Zenit earlier on Monday, barely threatened during a miserable first half and when Saido Berahino did eventually find the net just before the break, he was flagged offside.
After the break, Sterling again saw a tame effort kept out by Myhill, while Lambert curled a free-kick narrowly over the bar, and Berahino squandered a great chance at the other end.
Victory was sealed for City in the 59th minute when Kompany escaped Lescott to head Silva's corner beyond Myhill from a tight angle.