Paul Pogba halved the deficit from the penalty spot in the 87th minute though, before Victor Lindelof came up with a 92nd-minute leveller for the Red Devils, who are now just two points off fourth-placed Chelsea in the Premier League table.
Solskjaer made five changes to the team that started against Arsenal in the FA Cup on Friday night, with David de Gea, Phil Jones, Andreas Pereira, Juan Mata and Marcus Rashford all coming into the side.
Burnley's team, meanwhile, showed seven changes from the one that started their 5-0 defeat to Manchester City in the FA Cup, with only James Tarkowski, Charlie Taylor, Dwight McNeil and Jeff Hendrick keeping their spots.
Burnley had to weather an early storm at Old Trafford, with Pereira and Rashford both seeing attempts blocked in the opening five minutes as the home side forced the issue.
The visitors actually had the game's first shot on target in the eighth minute, but Wood's attempt from distance never threatened to cause De Gea problems.
Rashford had a golden chance to make the breakthrough one minute later after Lukaku found his teammate inside the Burnley box, but the in-form attacker somehow poked wide of the post.
McNeil then fired a long-range attempt over the crossbar down the other end, before Barnes had a shot taken off his tie by Jones as the visitors continued to take the game to United.
Mata started to find space inside the Burnley box as the match approached its 30-minute mark, and only a last-ditch challenge from Bardsley prevented the Spaniard from having a fine chance to make the breakthrough.
Pogba's long-range passing remained a feature of United's first-half work, but too many balls were overhit in the final third and Burnley were fairly comfortable as the first period approached its latter stages.
The visitors actually had the ball in the back of the United net in the 33rd minute when Wood lifted over De Gea, but the offside flag was correctly raised.
Luke Shaw had a powerful strike deflected behind for a corner in the 34th minute, before Pogba had a goal chalked off for offside four minutes later as the score remained level at Old Trafford.
Mata had the final attempt of the first period from distance, but the Spaniard's low strike was always moving wide of Heaton's post and the two teams headed down the tunnel at 0-0.
Both teams had chances in the early stages of the second period, but Barnes just could not pounce on a loose palm from De Gea, before Heaton kept out a low Mata strike down the other end of the field.
Heaton then had to save from Pogba in the 50th minute during an extremely open start to the second half of action.
Burnley made the breakthrough in the 51st minute when Cork robbed Pereira in a dangerous position before finding Barnes, who smashed into the roof of the net to stun Old Trafford.
Rashford brought a save from Heaton in the period that followed, before the England international missed the post with a low strike moments later as United desperately looked to level the scores.
Burnley were still committing players forward, however, and De Gea had to keep out a powerful McNeil effort in the 58th minute.
It was not Pereira's night at Old Trafford and it did not come as a surprise when the midfielder was withdrawn for Jesse Lingard in the 63rd minute, with Barnes's strike still separating the two teams.
Heaton kept out a close-range Lukaku attempt in the 67th minute as the pressure continued and that proved to be the Belgian's last piece of action as Alexis Sanchez followed Lingard onto the field.
Pogba headed a Sanchez cross over the Burnley goal in the 70th minute and the fact that Mata was booked shortly after for dissent was an indication of the home side's frustration on the night.
United continued to dominate the ball approaching the final 20 minutes, but Burnley were comfortable defending deep, with the likes of Tarkowski and Ben Mee outstanding on the night.
Burnley incredibly doubled their lead in the 81st minute when Wood headed an Ashley Westwood cross into the bottom corner of the net to silence the home supporters once again.
United halved the deficit in the 87th minute though when Pogba scored from the penalty spot after Hendrick had fouled Lingard, who made a huge difference coming off the bench.
Heaton made a stunning save to keep out a strike from Pogba in the final minute as the pressure continued, but United found an equaliser in the 92nd minute through Lindelof, who converted after Heaton had made another brilliant stop to deny Sanchez.
There was still time for the home side to look for a third, but the night ended 2-2 as United moved a step closer to fourth-placed Chelsea in the table, but dropped a potentially valuable two points.
MANCHESTER UNITED (4-3-3): De Gea; Young, Lindelof, Jones, Shaw; Matic, Pereira (Lingard 63'), Pogba; Mata, Rashford, Lukaku (Sanchez 68')
BURNLEY (4-4-2): Heaton; Bardsley, Tarkowski, Mee, Taylor; Hendrick, Cork, Westwood, McNeil (Gudmundsson 78'); Barnes, Wood