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Jack Grealish shot Villa into the lead with a superb solo effort midway through the first half, but United levelled before the break when Marcus Rashford's header hit the post and came off Villa goalkeeper Tom Heaton into his own net.
Victor Lindelof then headed United in front, but the lead lasted just two minutes before Tyrone Mings secured a deserved point for Dean Smith's visitors.
Villa remain just three points behind United in 15th, but edge three points clear of the relegation zone.
Grealish's excellent season continued when he shot the visitors into the lead after 11 minutes.
Anwar El Ghazi burst down the right wing before crossing through the United area for Grealish to gather the ball on the left-hand side and take advantage of Andreas Pereira's decision to back off by curling a precise shot into the top corner of David de Gea's goal.
