Daniel Caligiuri took advantage of some slack defending from United to put Wolfsburg ahead after four minutes, but Mata won and converted a penalty to equalize before half time.
The Spanish midfielder then back-heeled a pass straight into the path of Chris Smalling, who slotted the ball past Diego Benaglio to seal the three points in the 53rd minute.
“We started off sloppy and with that goal you give yourself a big battle to get back into the game,” Smalling said. “But after that we created a lot of chances and I think it was a deserved victory.”
United looked anything but potential Champions League winners, but manager Louis van Gaal should be satisfied his team secured the victory against a disciplined and threatening German opponent.
“A lot of players are very tired because of the (accumulation) of a lot of matches,” Van Gaal said. “But we have to keep the ball better in the second half because that was our problem.”
Though United lost its opening match at PSV Eindhoven, all four teams in Group B are now on three points after the Dutch side’s 3-2 defeat to CSKA Moscow on Wednesday.
United attacked with intent from the off, but the hosts were stunned in the fourth minute when Wolfsburg took the lead.
Smalling, Morgan Schneiderlin and Daley Blind failed to dispossess Caligiuri in the right corner. He played the ball inside to Max Kruse, who returned the ball to the midfielder in the box and he stabbed the ball past David de Gea.
United looked for an immediate reply, but struggled to find one. Dante flung himself into the path of Mata’s curling strike and diverted the goal-bound shot over the bar.
Anthony Martial shrugged off his marker and negotiated his way past another before rifling a shot just wide.
Matteo Darmian put in a crucial block deep inside the home penalty area and United immediately went on the counter. Martial turned his marker and squared to the unmarked Wayne Rooney but he planted the ball into the crowd.
Finally the nerves eased inside Old Trafford when Mata cut in from the right and swung in a cross which was impeded by Caligiuri’s outstretched left arm. Referee Viktor Kassai pointed to the spot and Mata slotted the ball into the bottom left-hand corner after taking over penalty duties from Rooney.
“The manager made a list of penalty takers and I am the first right now, so I have confidence,” Mata said.
United had more composure about them in defense after Ashley Young came on for Valencia at half time.
Wolfsburg looked for another early goal, but luckily for United, Darmian was on hand to deny Bas Dost an easy tap in.
United took control of the game eight minutes into the second half. Smalling rolled the ball past Wolfsburg goalkeeper Benaglio, but Mata’s contribution was decisive.
The Spaniard flicked a wayward pass over the Wolfsburg back four and into Smalling’s path. The defender slid in and guided the ball past Benaglio with his right foot.
Bendtner fluffed a good chance just after entering the field and Wolfsburg continued to apply pressure, but United hung on for its first Champions League win under Van Gaal.