Ryan Giggs was given too much space in the centre of midfield and lifted a simple ball over the defence. Tevez was far more alert than the sleepy defence and reached Giggs' pass, scoring with a first-time flick past Karim Zaza.
More goals looked on the cards as chances fell for Jonny Evans, Nani and Rooney again, all in the opening 20 minutes.
AaB, to their credit, were happy to soak up the pressure - albeit unconvincingly - and then find the gaps left by the home side, with Thomas Augustinussen and Marek Saganowski missing the target early on.
But the goals were still a shock when they came, both of them against the run of play.
Nani had almost scored after being played in behind the AaB defence when, two minutes later, he gave away a free-kick.
The set-piece was swung in to the front post where Michael Jakobsen had made a clever run, untracked by the static United defence, and he was able to glance a diving header beyond Tomasz Kuszczak and into the far corner of the net.
The second goal, scored just before half-time, mirrored the first with John O'Shea missing an identical chance to Nani's before AaB went up to the other end to shock the home crowd.
Anders Due found the space to whip in a deep cross which Jeppe Curth nodded back across goal and, miraculously, past Kuszczak. It was a wonderful, curling effort which gave the match some much-needed spice.
It at least snapped United out of the rut they had casually worked their way into after their early dominance, and the second half began in familiarly promising fashion.
Seven minutes into the half, after Tevez had already missed a good headed chance, Rooney powered his way into the AaB area and poked a low drive past Zaza from 15 yards.
Yet, much to the frustration of the United bench, the reigning European champions couldn't push on to clinch the three points.
A Steve Olfers sliding tackle took the ball from substitute Paul Scholes' feet just as he looked ready to score, and Zaza made two good saves from Gary Neville and Olfers' deflected header.
In truth they were probably affected by events in the other Group E game, which allowed them to finish top, but they would have liked to have taken their chances and cruised through to the next phase.