The Spaniard had a quiet night on his first appearance for the Red Devils, who earned victory thanks to an early goal from the returning Van Persie and a fine hit from Young
Spain international Mata was given his first outing in a United shirt since completing his club-record 37.1million pound deal from Premier League rival Chelsea on Saturday, but it was Van Persie - making his first appearance since Dec. 10 due to a groin injury – who stole the show by notching the opener.
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Mata played his part in a lively United performance, and he was afforded a standing ovation by the Old Trafford crowd as he was withdrawn in the closing stages, with Ashley Young's superb strike rounding off a pleasing night for boss David Moyes.
Cardiff manager Ole Gunnar Solskjaer, who enjoyed an 11-year spell as a player at United, could not spoil the party against his former club - the Norwegian enduring an unhappy homecoming as he continues to search for his first Premier League victory in charge.
The home side controlled the tempo in the first half and deservedly moved ahead when Van Persie scored his fifth goal in his last six Premier League games with a close-range header.
United continued to impress and Young added a second with a fierce strike just short of the hour mark, as United surpassed 1,000 points at Old Trafford in the Premier League.
United utilized Mata behind Van Persie, while Wayne Rooney was on the substitutes' bench after a five-match layoff with a foot injury.
As well as welcoming back Solskjaer, there was another familiar face for United fans in the Cardiff lineup, as the club named former Old Trafford striker Fraizer Campbell in its starting XI.
The home side was rewarded for a bright start with the opener after six minutes.
Antonio Valencia headed Young's teasing delivery against the crossbar and, although David Marshall saved Van Persie's initial follow-up, the Dutchman nodded home the rebound.
Mata was at the heart of all United's attacking play but, despite their solid start, it took until the 34th minute for the champions to fashion another chance of note when Patrice Evra's clever backheel found the lively Van Persie, who flashed an effort across goal from an acute angle.
Cardiff could have punished the hosts when Craig Noone crossed from the right, only for his delivery to narrowly evade Campbell.
The visitors enjoyed a spell of pressure after the break and Declan John's cutback only just escaped the attentions of Campbell again.
United was rarely in danger, however, and Young made it 2-0 on 59 minutes when he received the ball on the left before cutting inside and skipping past two challenges to rifle into the far corner from 25 yards.
Shortly afterwards, Rooney replaced Van Persie - and he was soon among the action with a deflected effort sailing narrowly wide.
Valencia then saw Marshall tip his effort onto the post as United piled on the pressure.
Mata saw an effort deflected over before he was withdrawn with time running out, receiving the adulation of his new fans.
Rooney then curled wide from 20 yards in injury time but United had done enough.