Villa were leading 4-2 heading into stoppage time at Old Trafford before late strikes from Joshua Zirkzee and Patrick Dorgu rescued a point for the hosts.
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Speaking after his 200th Premier League match, Emery praised his players' overall performance but expressed frustration at their defensive lapses.
"It's difficult to accept this result because we were winning 4-2 in extra-time and how we conceded two goals is really painful," he told VillaTV.
The Spaniard acknowledged that his team showed resilience and controlled much of the second half but stressed the need to improve.
"There are very good things, but the last six minutes was really getting us upset," he added.