Two goals from Paul Pogba and Victor Lindelof secure a crucial point for Manchester United.
Manchester United has been winning all matches since Solskjaer took over the club from Jose Mourinho.
The Red Devils were close to their first defeat under the care of Solskjaer when Burnley took the lead in the second half from a penalty spot through Ashley Barnes.
Chris Wood doubled the lead in the 81st minute heading home Barnes cross piling the pressure for Manchester United to make a comeback. Burnley early goals could be a rare win following a 5-0 thrashing by Manchester City in the FA Cup.
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Five changes were made from the previous side restoring the first team line up. Burnley goalkeeper and there back four were good enough to deny Manchester United clear goal chances both in the first half and in the second half.
Manchester United poor choices
Manchester United care taker manager however made poor choices in the Tuesday game bringing in Andreas Perreira, Romelu Lukaku and Juan Mata.
Perreira was poor as the league began and was easily dispossessed by Jack Cork of Burnley. Lukaku failed to make an impact in the match with Rashford becoming less effective after he was pushed to the left position.
Mata who played behind Lukaku and Rashford looked slow in pace as he tried to pass the ball or give the Red Devils a goal.
The draw extends Burnley unbeaten run to five games and Manchester United to nine games in all competition.