The win moves the Blues to within four points of second-placed Manchester City in the Barclays Premier League but it was only a fantastic effort seven minutes from half-time that separated the two sides.
Norwich had decent spells of possession but were too blunted in attack to ever truly test Petr Cech.
David Luiz saw two long-range efforts go high and wide in the opening exchanges before Mata fired a decent opportunity into the side-netting and John Obi Mikel shot just over.
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.
Bunn was called into action twice at the start of the second half to keep out an inswinging Oscar corner and a Victor Moses shot with smart saves.
The weather conditions worsened during and Norwich started to enjoy more of the ball in attacking positions. They could still not force Cech into a meaningful save but the movement of Wes Hoolahan and Grant Holt started to cause problems.
It was the visitors who still looked the more threatening, though, and Moses fired just wide before Chelsea substitute Eden Hazard forced another decent save from Bunn with only a couple of minutes remaining.
Hazard conceded a foul soon after which saw the resulting free-kick almost hooked into his own goal by Ashley Cole before Sebastien Bassong headed the corner over the bar as Norwich finished empty-handed.