United held off the reigning European champions with a battling defensive display before taking the lead through Nemanja Vidic's excellent 58th-minute header.
Bayern had failed to make their possession count, but Bastian Schweinsteiger's lethal finish for the equaliser still makes the German side strong favourites to go through.
United will at least not have to face Schweinsteiger in the second leg after the Bayern midfielder was sent off near the final whistle for a second booking after bringing down Wayne Rooney. Javi Martinez, too, will be suspended after he was booked.
"The players played really well so we're delighted but just a bit disappointed that we conceded in the end," said Moyes. "There was a real determination to make sure we did well, the players realised we were playing a really good side tonight.
"We know we've got to score a goal in the second leg. But we know we've given ourselves a great opportunity."
While there have been some historic encounters between these two sides, this match will be some way down the list. United fans could at least be grateful that this was not a home humiliation of the sort Liverpool and Manchester City have recently inflicted on David Moyes' side; given their record this season it looked a distinct threat from all-conquering Bayern.
The first half was a one-sided affair in favour of the Germans in terms of territory and possession. For all of that though, it was United, defending the 18-yard line in numbers and with tenacity, who managed to contrive the best chances.
Danny Welbeck promised some early joy when he rammed home a fierce strike from the edge of the box, but the whistle had already gone for dangerous play for a high foot, much to United's fury,
That was but a brief interlude, however, in the wave after wave of Bayern attacks, orchestrated by their wing wonders Arjen Robben and Franck Ribery. Robben curved an effort a yard past the post, then saw another 20-yard strike tipped around the post by David de Gea.
United's passing was pedestrian to the point of embarrassing, and Old Trafford's growing frustration with Marouane Fellaini came close to spilling over as the tall Belgian struggled with his passing and even found himself out-jumped for headers that he seemed clear favourite to win.
For all Bayern's superiority, there was always the suspicion that Jerome Boateng was a vulnerable point of weakness to Welbeck's pace. The England striker had already hit another firm shot straight at Manuel Neuer before Rooney played a through-ball, Boateng stumbled and Welbeck raced in on goal. Neuer advanced and read Welbeck's mind as he attempted the chip, leaving Rooney livid and giving his team-mate a blast of his own hair dryer.
Moyes had put his faith in the wise old head of Ryan Giggs in his starting line-up, the last survivor of the famous 1999 final triumph over Bayern and playing in his 141st Champions League match, one short of Raul's record.
But this old head was on old legs and after a mainly forgettable first half for the 40-year-old, Moyes made a predictable change, bringing on Shinji Kagawa in his place.
Almost immediately, United looked more threatening, with Welbeck and Rooney having greater support in attack and testing the Bayern defence. A couple of good moves saw United force a corner, which Rooney delivered with perfect pace and accuracy for Vidic to guide a wonderful header into the corner in the 58th minute.
Pep Guardiola responded by sending on Mario Mandzukic for Thomas Mueller, and the change paid dividends in the 66th minute as the Croatian striker headed back Rafinha's cross for Schweinsteiger to finish emphatically into the top corner.
Old Trafford, which had been re-energised by Vidic's goal, was suddenly deflated and it was Bayern who finished the stronger, with Moyes' men grateful that with two swerving efforts Robben - who broke United's hearts at this ground in the 2010 quarter-final - could not this time find the final flourish.
"It was a tough game," said Vidic. "One mistake and we were punished. They're going to have a lot of possession at their ground so we have to keep our shape. We can hurt them."