With the lead already at 332 runs going into the third morning, Sussex looked to get a move on with half an eye on having Sunday off.
The visitors batted for just over an hour, with spinners Felix Organ and Andrew Neal bowling all but two overs of the session, and in that time they racked up 103 runs before declaring on 319-9, leaving Hampshire the almost impossible task of chasing 436 to win.
Neal had Jack Leaning, who made 78 in the first innings, out early on for 12 before Haines was trapped lbw by Organ, looking to sweep to the short boundary, for a magnificent 130.
The wickets gave Jack Carson a chance to have some fun and he blasted 28 off just 16 deliveries – 24 of them coming the way of sixes in consecutive overs.
First-innings hero Jaydev Unadkat joined the party as Sussex decided they had tortured the hosts enough and skipper Haines called them in at 11:37 BST.
The Indian left-armer then struck with his third ball, hitting Organ on the pads after Nick Gubbins had been dropped at second slip the delivery before.
Price then had Toby Albert plumb in front before bookending his second over with the wickets of Tom Prest and Joe Weatherley for ducks, leaving Hampshire reeling at 3-4, with loud jeers from the crowd when a single brought the runs to level with wickets on the scoreboard.
After being 26-4 at lunch, Hampshire were soon 29-5 when Ben Brown was caught behind, but Gubbins remained strong, having had a life first ball.
He reached a battling half-century, with minimal celebration, and put on 85 with 18-year-old Ben Mayes for the sixth wicket, but both were dismissed in quick succession by the spin of Carson.
Neal and James Fuller battled through to tea, playing with the fluency that the top order would have liked, to guide Hampshire to the break at 192-7.
Tom Clark pulled off another screamer, to back up his one-handed grab from day two, to remove Fuller for 39 as Neal moved to his maiden Championship fifty.
But the job was done for Sussex when Neal and Kyle Abbott were pinned lbw by Price to give him his fifth career five-wicket haul.
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