It looked like a goalkeeper error by Robin Olsen, who let a header from Scott McTominay squirm past him in the 70th minute, would gift United a victory at Old Trafford.
However, in the last play of the game, a free kick from Lucas Digne was glanced on and Calvert-Lewin controlled the ball before prodding it past David De Gea.
United is two points behind City, which has two games in hand — the first coming on Sunday at Liverpool.
United led 2-0 at halftime thanks to Edinson Cavani’s header and a nonchalant strike by Bruno Fernandes, who floated a long-range shot over Olsen and into the corner after producing a dummy to create some space outside the area.
Everton responded with two goals in a four-minute span early in the second half, through Abdoulaye Doucoure and James Rodriguez.
Calvert-Lewin’s goal marked a remarkable end to a night that started with both sides paying their respects on the 63rd anniversary of the Munich air disaster.
It ended in a scoreline that brought back memories of a 4-4 draw between United and Everton at Old Trafford in 2012 that helped City win the league that season.
“We played some good football in the second half but conceded three goals from three shots on target. When you do that it is disappointing,” said United manager Ole Gunnar Solskjaer, whose team has conceded 30 goals — the highest number in the league’s top 10.
“We reacted well after their two goals but then why was there four minutes added in? We had to get the ball in the corner and see it out. Poor goals to concede.”
Everton manager Carlo Ancelotti said it felt “like a win.”
“We competed against one of the best in the Premier League,” he said. “We are not at the same level but we can fight and compete with our qualities. I am proud of my team.”
Soucek sent off as West Ham held 0-0 by Fulham in EPL
West Ham top scorer Tomas Soucek was sent off in second-half stoppage time in a 0-0 draw with relegation-threatened Fulham in the English Premier League on Saturday.
West Ham defender Vladimir Coufal struck the crossbar with a header in a rare opening for the visitors, who would have jumped into the top four — at least for one night — with a win at Craven Cottage.
They stayed in fifth place, a point behind Liverpool.
Fulham bounced back from a lacklustre 2-0 loss to Leicester in midweek and was able to subdue in-form West Ham striker Michail Antonio, but could not break the deadlock despite forcing some late half-chances.
The draw means only Brighton have had more draws this season than Fulham’s nine, and left Scott Parker’s side in third-to-last place, eight points from safety.
Burnley, Brighton draw 1-1 to inch clear of EPL bottom 3
Johann Berg Gudmundsson made amends for an earlier mistake by scoring the equalizer for Burnley to match Brighton 1-1 as both sides stayed clear from the English Premier League relegation zone on Saturday.
Lewis Dunk’s towering header from a corner sailed into the net in the 36th minute despite the presence of Gudmundsson at the back post, with the Iceland winger seemingly losing his footing in the pouring rain at the crucial moment.
However, Gudmundsson’s strike eight minutes after halftime ensured an eighth draw in the most recent 11 meetings between the teams.
Burnley remained 17th and stayed eight points clear of the bottom three, after third-to-last Fulham drew with West Ham 0-0 later Saturday.
Premier League: Man United concede in stoppage time, draw with Everton 3-3 added by News89 Team on
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