Hampshire had been put on the ropes by four quick wickets in the afternoon session but the pair batted for 25 overs to save the game for the hosts.
Warwickshire were in a great position to attempt to force a result after Ian Bell further helped his chances of an England Test recall by scoring miserly-turned-adventurous 174.
The Warwickshire captain notched up 44 morning runs with a masterclass of field piercing shots – showcasing reverse sweeps, cover drives and the rest of his repertoire.
He reached 150 in 268 balls and looked to attack the ball to keep the score moving and put pressure on Hampshire.
At the other end the visitors lost their first wicket of the day, Rikki Clarke on the wrong end of an Adam Wheater stunner, as the keeper dove low to his right to grab an outside edge.
Bell was the next to fall as he tried one shot too many and skied Ryan McLaren to James Vince at mid-on.
Spinner Dawson had bowled brilliantly on day three and finally got the wicket his tight-fisted spell deserved as Jeetan Patel’s pole was knocked back.
Boyd Rankin was the last to depart as he copied his captain and lofted one to Vince – Warwickshire ending on 360, with a lead of 158.
Hampshire wobbled in their effort to save the game, as Warwickshire’s bowling attack shared the wickets.
Keith Barker got two while Clarke, Patel and Rankin all grabbed a scalp each, with the latter picking up his sixth of the match.
Fast bowler Barker was the first to strike as Michael Carberry edged to Varun Chopra at first slip, before Sam Hain pulled off a close catch to get rid of Tom Alsop – giving Rankin his wicket.
Spinner Patel caused problems throughout the session, with regualar oohs and aahs when the ball gripped, and got his reward when he had Will Smith leg before.
The hosts found themselves in further trouble when skipper Vince was also out lbw to Clarke.
Barker ended the game with match figures of 7-83 – his last victim a hapless Sean Ervine, who stuck his bat out at a wide one unsure of his favoured final outcome.
But keeper Wheater and all-rounder Dawson took over and guided Hampshire into a lead.
And hands were finally shaken just before 6pm, after Dawson completed his half-century to salvage a draw – Hampshire nabbing what could be a useful nine points, Warwickshire collecting 12 to kick-start their title bid.
Warwickshire coach Dougie Brown said: “It was disappointing to lose nearly four sessions due to bad weather but we can’t do anything about that.
“What we can do is influence the game in a positive way in the time we do have remaining in the game. I think we gave ourselves a good opportunity and if things had gone for us a bit more, not that they went against us, we created that opportunity for ourselves.
“We can’t ask for any more than that from the side.
Hampshire director of cricket Giles White was delighted with Dawson and Wheater, adding: “It was a very important partnership at that stage of the game. We were under pressure at that moment and they both came through and showed some character, batted well and made good decisions and saw us to the draw.
“We will take the points and move on to Headingley.
“We knew it was going to be tough. It is well known on the circuit that you have Barker bowling his left-arm seam and gives the foot holes Patel bowls well within the last innings. We had to contend with that today from a skilful bowler.
“With Reece injured a couple of quick wickets and we would have been under pressure. It was good to see them hold firm.”