Barnes gave Burnley the lead in the third minute as he reacted quickest to a loose ball from a free-kick inside United’s area and tucked it home.
In the 36th minute, Defour doubled Burnley’s lead as he curled a free-kick into the top-corner.
Lingard pulled a goal back for United in the 53rd minute following a brilliant back-heel into the bottom corner to make it 2-1.
And in the 92nd minute Lingard drew United level after pouncing on the ball and slamming it into the net to rescue a point for his side.
United have now gone three straight games without a win in all competitions.
They remain second in the league table on 43 points, 12 points from leaders Manchester City, who have a game in hand.
At Stamford Bridge, Chelsea returned to winning ways following their 2-0 home win against Brighton and Hove Albion.
Super Eagles star Victor Moses was in action for Chelsea and played for 90 minutes.
After a goalless first-half, Chelsea eventually took the lead on 46 minutes as Alvaro Morata headed in Cesar Azpilicuerta’s cross.
In the 60th minute, Chelsea went 2-0 up this time from Marcos Alonso who headed in Cesc Fabregas’ corner.
The win means Chelsea maintain their third spot on 42 points.
And at Vicarage Road, Watford came from 1-0 down to beat Leicester City 2-1.
Nigeria midfielder Wilfred Ndidi was in action for Leicester while his Super Eagles teammates Ahmed Musa and Kelechi Iheanacho were left out of the Foxes squad.
Leicester went 1-0 up in the 37th minute thanks to Riyad Mahrez who headed in at the far post from Marc Albrighton’s cross.
On the stroke of half-time, Molla Wague pounced on a loose ball at the far post and smashed it into the back of the net.
Watford went in front in the 65th minute off Abdoulaye Dacoure’s shot from a tight angle which crept over the line after reflecting off Kasper Schmeichel to make it 2-1 to Watford.
Leicester are eighth on 27 points and Watford 10th on 25 points.
In the early kick-off at Wembley Stadium, Harry Kane netted a hat-trick as Tottenham Hotspur thrashed Southampton 5-2.
Chelsea 2 vs 0 Brighton
Watford 2 vs 1 Leicester
Man United 2 vs 2 Burnley
Bournemouth 3 vs 3 West Ham
Huddersfield 1 vs 1 Stoke
West Brom 0 vs 0 Everton