However, Daley Blind grabbed another equaliser with three minutes remaining to give the Red Devils a share of the points away from home.
After the match, Dawson was in an optimistic frame of mind after helping the Baggies record their third draw in eight Premier League matches so far this campaign.
"It was a great team performance and we did well to get the second goal," he told Sky Sports "It's a shame we couldn't hold on in the end.
"First time playing against United, it was difficult. They've got so many great players, but I think the team played really well tonight.
"We've got to take that into the next game. We'll take the positives out of tonight."
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