Leaders Arsenal, meanwhile, blew an opportunity to move four points clear after drawing 2-2 at Southampton, but Liverpool tightened their grip on fourth place with a superb 4-0 defeat of local rivals Everton.
Mata started at Old Trafford, three days after making a £37.1m move from Chelsea, as former United striker Ole Gunnar Solskjaer returned to his old stomping ground as Cardiff manager.
It took last season’s top scorer Robin van Persie just six minutes to announce his return from injury when he broke the deadlock heading in at the second attempt after his initial header was parried by goalkeeper David Marshall.
Ashley Young added a fine second in the 59th minute, cutting in from the left flank and firing an unstoppable shot into the bottom-right corner.
Down at St Mary’s, Arsenal missed a golden opportunity to move four points clear of Man City, ahead of City’s match with Spurs on Wednesday.
Arsenal were really poor in the first half and were lucky to go into half-time trailing just to Jose Fonte’s goal.
After a rollocking from Arsene Wenger, Arsenal got back on level terms through Olivier Giroud and then went ahead from Santi Cazorla. But Arsenal were only in the lead for minutes as Adam Lallana tied things up two minutes later.
In the big merseyside derby, Liverpool took Everton apart 4-0 at Anfield to move three points clear in fourth. Steven Gerrard’s 21st-minute header from Luis Suarez’s corner broke the deadlock and two quick goals from Daniel Sturridge before half-time ended the contest. Luis Suarez raced clear to add a fourth early in the second half, before Sturridge squandered an opportunity to complete his hat-trick by blazing a penalty over the crossbar.
Elsewhere, goals from Jonjo Shelvey and Chico Flores gave Swansea City a 2-0 win at home to relegation-threatened Fulham, while improving Crystal Palace climbed to 14th after an early Jason Puncheon goal saw them beat Hull City 1-0.