Dawson, who had scored 61 in the first innings, got his side out of a sticky position at 103-6 when chasing 197 with a boundary-filled 67 – with a crucial 67-run stand with Keith Barker.
The victory – a fifth in seven Championship fixtures in 2022 - was Hampshire’s first over Yorkshire at home in the Championship since 2008.
Hampshire claimed 22 points to keep pressure on Surrey, with the gap at the top of Division One just three points. Yorkshire, who had scored over 400 in their first innings, take away six points from the topsy-turvy match to drop to fourth.
Barker had led a third evening assault on Yorkshire’s top order, with the left-armer claiming three of five wickets to leave the visitors on 101 overnight. Hampshire wanted a quick repeat in the morning session to set up an easy chase.
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It started perfectly as only two runs had been added before Harry Duke edged Kyle Abbott onto his stumps with the seventh ball of the day.
Dom Bess was then lbw to the South African, with Jordan Thompson driving Brad Wheal to wide second slip two balls later.
At that point, Yorkshire’s lead was just 142 with a bountiful number of overs but Dominic Drakes and Matthew Revis combined to add 44 runs while eating up 68 minutes. Both soaked up deliveries – typified by four maidens on the bounce – with Drakes’ pulled six a rare moment of aggression.
Ian Holland eventually broke the partnership with the first ball following lunch – Drakes driving to first slip – before Steven Patterson had his off stump removed by Wheal, with Revis unbeaten on 28.
Felix Organ began the 59-over chase with two cut boundaries but he failed to score another run before he guided a back-of-a-length delivery from Thompson to third slip. Thompson then pinned Holland in front to leave Hampshire 23-2.
Nick Gubbins and James Vince had rescued a similar situation in the first innings with an 82-run partnership, and this time around they showcased their famed shot-making traits. Both edged over the slips but Vince’s front foot pull and Gubbins’ all-around play caught the eye.
Both struck sixes – Gubbins on the pull, Vince with a leg-side jab – to keep up with, and get ahead of, the required rate.
The partnership ended on 55 after 59 balls when Gubbins was struck on the pads by Patterson while attempting a sweep before Vince feathered Bess behind six balls later. Hampshire still 117 runs away from their target.
Only 14 more runs had been added when Bess found some good turn and bounce to develop an edge to slip before Aneurin Donald was bowled through the gate by Patterson just nine more runs down the line.
Dawson and Barker took a risk-free approach and ticked off the runs at a sullen rate.
Any pressure was relieved in the 38th over when Dawson drove, advanced over long-on and swept his way to a four-six-four combination - with 16 runs coming from the over. He reached his 54-ball half-century with a boundary in the following over, which also included the 50 stand with Barker.
Even as Barker was strangled down the leg side for 21 with 27 runs still needed, Hampshire had things under control with Dawson unfurling three glorious cover drives.
Dawson hooked to long leg with seven required but James Fuller and Abbott took Hampshire over the line, with Fuller's uppercut completing the chase with 13.4 overs to spare.
Dawson said: ‘That was a great advert for four-day cricket on a really good wicket and we are really happy to get over the line.
‘Any total over 150 in the fourth innings is always tricky but we thought the pitch was really good and luckily we bat deep.
‘I feel like I've been in decent form all year without big scores and I'm really happy with that innings. You always feel pressure every time you bat but luckily today I was able to get us the win.
‘I was in a bit of a daze, just watching the ball and reacting to it and luckily it paid off.
‘It is a brilliant comeback and shows the character in the group. It is probably one of the best wins I've been involved in with Hampshire in four-day cricket.’
Vince added: ‘It was a bit nervy at the end. To come back from losing the toss and them getting 400 is outstanding. The changing room is buzzing.
‘We were happy chasing anything around 200. The run rate was never really an issue. It was a bit frantic at times but the way Daws held it together under immense pressure was outstanding.
‘There is still a lot of cricket to be played but when you see Surrey are winning you feel the pressure to stay up with them. We haven't made any ground but we have stayed right up there.’
Hampshire’s next Championship game starts on June 26 against Essex at Chelmsford.
Yorkshire 428 & 178 lost to Hampshire 410 & 198/8 by two wickets