Leaders Arsenal, meanwhile, spurned 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 derby 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.
Last season's top scorer Robin van Persie was also included in United's starting line-up, after seven weeks out with a thigh strain, and the Dutch striker broke the deadlock in the sixth minute.
Mata found Ashley Young with a raking pass and when his cross was headed against the bar by Antonio Valencia, Van Persie headed in at the second attempt after his initial header was parried by goalkeeper David Marshall.
Young added a fine second in the 59th minute, cutting in from the left flank and arrowing a drive into the bottom-right corner, before Wayne Rooney made his return from injury as a substitute for Van Persie.
United, who had lost three of their previous five home league games, remain six points below the UEFA Champions League places, but they inched to within 12 points of Arsenal after the London club's slip-up at St Mary's.
Arsenal slip at St Mary's, Liverpool in memorable derby win
Bidding to move four points clear of second-place Manchester City, Arsenal fell behind in the 21st minute when centre-back Jose Fonte headed in a left-wing cross from Luke Shaw.
The visitors sprang to life early in the second half, equalising through Olivier Giroud's deft back-heel and going ahead when Santi Cazorla drilled home, only for Adam Lallana to sweep in an equaliser two minutes later.
Arsenal's push for a late winner was undermined when midfielder Mathieu Flamini was sent off for a studs-up lunge at Morgan Schneiderlin.
Arsene Wenger's side are now at risk from City, who will take over at the top if they win at Tottenham Hotspur on Wednesday, while Chelsea could draw level with Arsenal if they win at home to West Ham United.
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.
Both sides finished with ten men at Carrow Road after Newcastle striker Loic Remy and Norwich midfielder Bradley Johnson were shown straight red cards following a late altercation.
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.