Antonio Valencia made United's 17th-minute opener at the Stadium of Light, getting space away from Patrick van Aanholt on the right edge of the box before firing in a low cross that Mata was able to tap home from four yards out.
But on the half-hour mark, Sebastian Larsson's corner was met with a firm downward header by £10million summer-signing Rodwell for the Sunderland equaliser, with Valencia at fault due to his lacklustre marking.
After the break, Ashley Young was booked for diving seconds after a Robin van Persie penalty appeal had been waved away by referee Martin Atkinson, and Sunderland goalkeeper Vito Mannone later ventured outside his area to tackle Young after he was played into space by Danny Welbeck.
The Black Cats are 11th in the early table after starting the season with two draws, whlile United - who again played with three centre-backs and Young and Valencia as wing-backs - are 13th after Louis van Gaal collected his first point as manager.
Sunderland head coach Gus Poyet made two changes to the team which started the 2-2 draw at West Brom, as Santiago Vergini and Will Buckley replaced Valentin Roberge and Adam Johnson.
Van Persie was one of three changes to the United side which lost 2-1 at home to Swansea last weekend, with he, Valencia and Tom Cleverley replacing Javier Hernandez, Jesse Lingard and Ander Herrera.
United's fragile back three of Phil Jones, Chris Smalling and Tyler Blackett was further reduced in terms of experience when Smalling limped off before the break to be replaced by Michael Keane.
But it was not only at the back that the visitors struggled, with Sunderland clearly carrying the momentum of last season's two victories over United, and skipper Wayne Rooney and strike-partner Van Persie saw little of the ball where it mattered most.
Nevertheless, it was United who took the lead, although they did so at a time when the breakthrough looked more likely to come at the other end.
Young had already put his defence under pressure with an aimless crossfield pass and Lee Cattermole unleashed two long-range efforts when winger Buckley, making his full debut for the Black Cats, cut inside Blackett and cross for Connor Wickham, who wastefully scuffed his shot straight at keeper David de Gea.
Sunderland were made to pay within seconds as Valencia managed to ease past full-back van Aanholt and cross low to the far post, where Mata ran off the back of Larsson to score from close range.
The goal visibly settled United and they enjoyed a sustained period of possession as they threatened to take control.
However, they allowed their hosts back into the game on the half-hour when, after Buckley's direct play had earned his side a corner, Larsson delivered the resulting set-piece to the near post and Rodwell headed firmly home with Valencia having done little to impede his progress.
United's response was lukewarm, and there was little doubt that Poyet would have been the happier of the two managers as their respective teams left the pitch at the break.
Indeed, the home side might have taken the lead six minutes after the restart when Wickham was given time to work himself into a shooting position inside the box and Blackett somehow blocked his effort with De Gea beaten.
But Van Gaal's men gradually assumed control as they dominated possession, although with Sunderland defending deep, space was at a premium even with Rooney dropping ever deeper.
They almost found a way through with 62 minutes gone when Van Persie surged into the box only to be halted by Vergini's last-ditch challenge, and Young was booked for simulation after going over Wes Brown's leg as the pair close don the rebound.
Van Persie departed seconds later as he and Darren Fletcher were replaced by Welbeck and Adnan Januzaj, but Larsson failed to make the most of Wickham's fine 68th-minute through-ball as United lived dangerously once again.
Larsson passed up a good opportunity when he curled a free-kick well wide with 15 minutes remaining with Sunderland hunting all three points.
To that end, Poyet attempted to pile the pressure on the visitors' rearguard with the introduction of battering-ram frontman Jozy Altidore, although defensive confusion inside the Sunderland penalty area very nearly cost them on more than one occasion in a frenetic finish.
United travel to MK Dons in the Capital One Cup second round on Tuesday and then Burnley in the league on Saturday, while Sunderland go to Birmingham in the cup on Wednesday before heading for QPR next weekend.
Watch United's Capital One Cup clash at MK Dons live on Sky Sports 1HD on Tuesday night, with coverage underway from 7.30pm.