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.
Sterling had the opportunity to open his account for City in the first half as he was sent through one on one but Myhill denied the winger.
