The win moves the Blues to within four points of second-placed Manchester City in the Barclays Premier League table but it was only a fantastic effort seven minutes from half-time that separated the two sides.
Mata, who scored his sixth goal in eight games with aplomb, also missed a good opportunity with the scoreline 0-0 while Norwich had decent spells of possession but were too blunted in attack to ever truly test Petr Cech.
The Blues were almost ahead after 30 minutes as Luiz robbed Wes Hoolahan of the ball before picking out Mata, who shot into the side-netting when well placed.
The sustained spell of Chelsea pressure eventually paid off and it was Mata who opened the goalscoring in style. The Spain international cut in from the left before letting fly with an effort from 25 yards that flew past Mark Bunn.
A near-post corner at the start of the second half almost doubled Chelsea's advantage as Oscar's set-piece evaded everyone before Bunn reacted well to push the ball wide.
Chelsea substitute Eden Hazard forced another decent save from Bunn with only a couple of minutes remaining after being found by a deft flick from Fernando Torres, before receiving a caution moments later for a cynical clip of Martin's heels as the Norwich full-back looked to counter.
The resulting free-kick was almost hooked into his own goal by Ashley Cole before Sebastien Bassong headed the corner over Cech's bar.