Fernando Torres had an early chance for the hosts when he was played through on goal but the Spaniard dithered for too long, allowing Leon Barnett back to clear.
The Canaries couldn't hold on to their advantage for long, however, as Torres made amends for his earlier hesitancy by nodding in Branislav Ivanovic's delivery.
Chelsea then went ahead midway though the opening period when Frank Lampard beat John Ruddy after Torres's acrobatic effort was blocked.
Chelsea got their fourth 15 minutes from time when Ivanovic slammed home after Mata had managed to get his foot to the ball.
Defeat for Norwich sees them drop a place to 19th.