Blues fans booed substitutions made by the former Napoli boss during the game, before joining the visiting fans in chanting “you’re getting sacked in the morning” at the Italian.
Former Chelsea striker Sutton doesn’t believe Sarri will still be at the club by the time they face Malmo in their Europa League last-32 second leg on Thursday.
The defeat to United followed a 6-0 loss to Manchester City and 4-0 collapse against Bournemouth in the Premier League in recent weeks, while they face City again in the League Cup final on Sunday.
"This is Maurizio Sarri's last game at Chelsea,” he told the BBC.
“The Italian job is over for them. He's done. Sarri-ball is broken.
"He'll get the sack for numerous reasons - things he's said, that he can't motivate the players, his inability to be able to adapt and change.
"He won't survive tonight. Well he might until 9am tomorrow [Tuesday]."
"Why risk him taking charge of a game against Manchester City?"
After the League Cup final, Chelsea face a Premier League clash at home to Tottenham three days later.
Sarri’s side are currently sixth in the table, one point behind United in the fourth and final Champions League place.
