On-loan Swansea forward Oliver McBurnie had given the visitors the lead with a first-half header but Dawson shot home with 17 minutes remaining to leave the Tigers three points and two places above the drop zone, one point and one place better off than Barnsley.
The visitors had to replace Adam Jackson shortly before half-time after he suffered a serious-looking injury in a clash of heads with Hull’s Jon Toral, with the Barnsley player taken off on a stretcher after a stoppage of more than 10 minutes.
Hull had the better of the early exchanges and won two corners in quick succession but neither came to much, Adama Diomande heading straight at Nick Townsend from the second.
The home side were getting most success down the right but full-back Ola Aina’s low cross was easy for Townsend to gather.
Barnsley’s first chance fell to Mamadou Thiam after 13 minutes but he headed a cross from Dimitri Cavare well over following a quick free-kick routine.
Barnsley started to come back into the game and took the lead after 21 minutes as McBurnie grabbed his third goal in two matches, rising unmarked to power a header into the top right corner from a Brad Potts cross.
Hull decided to go route one but when Nouha Dicko crossed from the right no home player was able to get a telling touch.
Dicko flashed a shot across goal following an exchange of passes with Toral as the home side pressed in search of an equaliser.
Toral headed wide from a Jarrod Bowen cross and went down injured along with Jackson after a clash of heads as half-time approached, with the latter coming off worst.
Jackson had only recently returned from a knee ligament injury suffered against the Tigers in October. His club tweeted after the match that he had ‘regained consciousness quickly’ and was talking to medical staff.
Hull started the second half in determined mood and after winning three successive corners Jarrod Bowen’s inswinging free-kick was met at the far post by Angus MacDonald’s header from a tight angle, which Townsend kept out with his legs.
Bowen then headed at Townsend from a Max Clark cross as the Tigers maintained the pressure, Dicko shooting weakly on the turn from another opening set up by Clark.
Hull claimed the equaliser their pressure deserved with 73 minutes gone when Seb Larsson’s corner from the left was not properly cleared and the ball fell to Dawson who applied a low finish from the edge of the area.
Dawson just failed to get a clear header on a Bowen free-kick as time ran out and both sides were forced to settle for a point.
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