Currie had earlier contributed his own List A best 61 not out off 40 balls.
Yorkshire started the game well, Matt Milnes in fine form again as Hampshire slipped up having been inserted.
Seamer Milnes, having claimed a career best 7-38 in last Sunday’s group-stage win over Sussex at Hove, claimed the first three wickets, including forcing visiting captain Nick Gubbins to play on with his third ball in the second over.
He then removed Fletcha Middleton and Ali Orr before George Hill’s seam also forced Ben Brown to play on.
Dawson came in at 53-3 in the 12th over and offered a sharp return catch to Ben Cliff on six, clearly a key moment.
The 35-year-old shared 89 for the fifth wicket with 17-year-old Ben Mayes, whose 37 helped to turn the tide, before sixth-wicket partner Currie pressed the accelerator.
Shortly after Dawson reached his fourth List A century off 103 balls, Currie’s maiden 50 came in 35 as Hampshire pushed on from 180-5 after 40 overs.
Dawson finished with seven sixes and hit strongly down the ground and over cover and long-on, while both he and Currie improvised as they shared 136 inside the last 13 overs of the innings – with 75 runs coming off the last five.
Kyle Abbott and Brad Wheal then bowled very tidily with the new ball, restricting Adam Lyth and Imam to 43-0 in the 13th over when the rain arrived.
A half-hour delay was followed by Lyth edging the second ball back behind off Eddie Jack but Ul-Haq, who is leaving for national team commitments after this game, then calmly advanced the hosts to 98-1 after 20 overs with a 52-ball 50.
Strong off his legs, the left-hander then united with Will Luxton to share 99.
Luxton pulled a couple of sixes, including one the first ball back after a second half-hour rain break, but he chopped on to Jack with the second, falling for 30 and leaving his side 146-2 in the 28th over.
James Wharton holed out to Currie shortly afterwards before Ul-Haq reached his fourth ton of this season’s campaign off 96 balls.
But he was beaten Jack’s direct hit and Fin Bean followed as Yorkshire lost a defining three wickets for 10 inside two overs to slip to 171-5, still needing 83 more.
Yorkshire then lost Hill and Harry Duke in successive balls to Brad Wheal and Dawson and, seven wickets down, needed 47 off 4.5 overs.
From there, they subsided, with Dawson striking again to remove Milnes and finish with 2-55 to add to his century.
ECB Reporters’ Network supported by Rothesay.
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