The Red Devils were facing just a second defeat at Old Trafford in 43 outings, and a first to the Clarets here since 1962, before Lingard bagged goals at the start and end of the second half.
Sean Dyche's men had been well on course for another dream result in a season to remember, leaving them in the hunt for a top-four finish, but they could not hold off their opponents and eventually came undone late on.
Burnley, in their highest Christmas-time position since 1973 heading into this fixture, had a lead to hold on to with just three minutes on the clock through a close-range finish from Ashley Barnes.
From their second attacking move of the afternoon, the Clarets very nearly had a second goal as Scott Arfield helped on a Phil Bardsley cross, which looked to have the beating of David de Gea but not the crossbar.
Either side of that chance, though, the home side were asking all the questions as Burnley - boasting the fourth-best defensive record in the division prior to today - continued to stand firm at the back.
Ben Mee produced a big block to prevent Juan Mata from scoring and goalkeeper Nick Pope was required to keep out Luke Shaw and Paul Pogba, the latter coming close through his headed attempt from a corner.
United were left stunned soon after, however, as Steven Defour expertly sent a free kick flying past De Gea's reach from 30 yards to double his side's tally.
In response, Pogba curled a shot wide and Marcus Rashford was denied on the line by Mee, as the Red Devils were left frustrated.
Jose Mourinho made a double change at the midway point in an attempt to revive his side's hopes, and it worked well as one of the fresh faces dragged the hosts right back into the game.
Having missed a glorious chance from five yards out, which hit Pope's face and then the crossbar, Lingard picked out the bottom corner with a smart backheeled attempt soon after.
A free kick was then sent narrowly over the crossbar by Pogba as United continued to dominate proceedings, without being able to seriously test Pope.
As the match entered the final 10 minutes there was no real change to the flow of the game, with the Red Devils bossing possession but lacking creativity, failing to muster another shot on goal until Lingard's late leveller.
The goal was perhaps harsh on Burnley, who defended well throughout, as the England international pounced after the ball deflected off Mee's leg to pick out the bottom corner and earn his side a share of the spoils.
Back-to-back draws leave United in second place in the Premier League, now just one point ahead of Chelsea, while Burnley are also non-movers in seventh.
Manchester United: De Gea, Young, Jones, Rojo (Mkhitaryan 46'), Shaw, Mata, Pogba, Matic, Rashford, Ibrahimovic (Lingard 46'), Lukaku
Burnley: Pope, Bardsley, Mee, Long, Taylor, Gudmundsson, Defour (Vokes 67'), Cork, Arfield, Hendrick, Barnes (Walters 81')