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United substitute Benjamin Šeško appeared to have secured all three points when he put the visitors ahead in the 88th minute, but Everton substitute Ainsley Maitland-Niles rescued a point with a superb strike from outside the box in the dying moments.
Manchester United, who finished third last season and consequently returned to the Champions League, opened their campaign with a defeat away to Hull City, before recovering in the second round with a convincing 5-2 win over Ipswich Town.
The visitors looked set for their second win after Šeško responded to Tyrique George’s 83rd-minute equaliser by restoring United’s lead four minutes later.
However, Maitland-Niles produced a stunning long-range effort to deny United victory and spare Everton their first defeat of the season.
“We felt we had won. We fought really hard and managed to come away with a point in the end,” Maitland-Niles told British broadcaster Sky Sports.
“The fans were behind us, and we fought until the very last moment. That’s why you get results like this.”