Dawson struck for the second successive home game, showing instincts of a striker to fire home from close in after Dieumerci Mbokani did superbly to nod down Robert Snodgrass's free-kick into his path.
It was just reward for City, who scored character after a desperately poor showing in the first period when they created nothing.
West Brom, seeking a third successive top-flight win for the first time since September 2012, had led thanks to a 34th-minute header from Gareth McAuley.
But perhaps inspired by some rousing interval words from Mike Phelan, Hull looked a different proposition on the restart.
Phelan said: "I thought we had the better of the second half but we could have conceded at the death and I could be sitting here talking about a defeat.
"But we got away with it - we're still in virgin territory a bit and we have to learn to concentrate for 96 minutes of the match because it is never over."
Opposition boss Tony Pulis admitted Hull deserved their equaliser.
Pulis added: "In the second half you have to give Hull credit - they started well and put us on the back foot and had a couple of opportunities.
"I thought we'd weathered that but obviously they get the goal, but the great thing is this group of lads know how to keep working."
A desperately low quality first half of football possessed little to set the pulses racing, particularly from a home perspective - with the Tigers bringing next to the nothing to the table.
Passes were overhit, wrong options were taken and the tempo was super-slow, with West Brom's back four remaining untroubled against a side who looked bereft of confidence for large spells.
At the other end, Albion were hardly free-flowing themselves, although they displayed the convinction on show and forged a breakthrough nine minutes before the interval.
An inswinging corner from the left from Matt Phillips was attacked with intent by McAuley, who powerfully headed home and took advantage of some meek Hull defending.
A carbon-copy almost yielded an identikit second moments before the break with Craig Dawson almost getting on the end of another well-delivered Phillips corner.
Aside from that, little else happened with any pockets of build-up play coming from the visitors, with James Morrison and Dawson off target with efforts.
An interval change saw Ryan Mason make way for Adama Diomande as Mike Phelan sought to find some badly-needed impetus after a waste of an opening half.
The tempo at least, if not the quality, was better from the Tigers on the restart with ironic cheers greeting their first shot on fifty minutes, with Robert Snodgrass's low effort held by Ben Foster before Jake Livermore fired over.
Livermore then forced Foster into belated action which an effort which almost yielded a leveller, with the ex-England keeper expertly saving his low shot after some neat interplay by the hosts and some quality at last.
Next to threaten was Sam Clucas, who headed wide, with the hosts starting to finally believe in themselves.
A good opportunity was afforded the hosts with McAuley was cautioned for a foul on Diemerci Mbokani, with Foster showing his mettle to turn away Snodgrass' low angled free-kick, which was destined for the corner.
After a spell of home pressure, the visitors - who struggled to get going in the second period - looked to have regrouped, but Hull had other ideas.
A deep free-kick on the left from Snodgrass was met by Mbokani, with Dawson reacting quickest to his knock-down to fire home an equaliser from close range 18 minutes from time.
Hull's endeavour on the restart was worthy of reward and the question remained if they could push for a winner.
That did not transpire and it was Albion who went closest to a late winner, with David Marshall rescuing the Tigers by making a fine parry to keep out an excellent free-kick from substitute Nacer Chadli.
Albion continued to push for a winner at the death and Solamon Rondon spurned a great chance to win in stoppage time, but headed James McClean's centre wide - with Hull holding out to secure a deserved point after a much improved second half showing.
Hull City: Marshall; Elmohamady, Dawson, Davies, Tymon (Robertson 90); Snodgrass, Clucas, Livermore, Henriksen (Meyler 89); Mason (Diomande 46), Mbokani. Substitutes unused: Jakupovic, Maguire, Huddlestone, Bowen.
West Brom: Foster; McAuley, Evans, Dawson, Nyom; Phillips (Robson-Kanu 83), Yacob, Fletcher (McLean 64), Brunt; Morrison (Chadli 79), Rondon. Substitutes unused: Palmer, Olsson, Gardner, Galloway.
Referee: P Tierney (Lancashire).
Attendance: 18,086 (1,045 West Brom supporters).