Vince hit 33 boundaries in his career-best 240, while Will Smith ended 151 not out as Essex declared on 440-3
The pair put on a record stand for Hampshire against Essex of 387 as they set the visitors a massive 606 to win.
Essex were all out for 135 as the hosts equalled the second-biggest victory by a runs margin in Championship history.
It fell 13 short of the record of 483 runs, set by Surrey against Leicestershire at The Oval in 2002.
But it did beat Hampshire’s previous first-class best victory of 361 runsagainst Cambridge University at Fenner’s in 1984, while it was Essex’s heaviest-ever defeat in terms of runs.
South Africa international Kyle Abbott (3-23) impressed with the ball again and finished with match figures of 8-67.
Smith, whose earlier unbeaten ton was his highest score for Hampshire, took the final wicket to fall when he had Tymal Mills caught by Vince to cap a memorable day for both players.