Dawson smashed 52 runs off 35 balls in a total of 6-173 on Tuesday (Wednesday AEST) to help steady the ship after Hampshire had fallen to 4-42.
A flurry of sixes from the lower order made an important late contribution for the home side to set Sussex a formidable target.
Australian allrounder Cartwright certainly did his bit with some lusty late hitting, blasting two sixes and a four in his unbeaten 24 off 17 balls.
Cartwright and Dawson, who know each well, having been teammates with BBL side Melbourne Stars, put together a partnership of 28 runs off 3.1 overs, before the Englishman lost his wicket in the 17th over.
The Aussie then shared an unbroken 37-run stand with James Fuller, who remained 19 not out off just nine deliveries that also featured two sixes.
Dawson was then the pick of the bowlers and finished with 3-20 as Sussex were bowled out for 144 with 15 balls to spare.
Australia's Daniel Hughes got the visitors off to a bright start, the opener making 31 off 22 balls (two fours, two sixes) and sharing in a first-wicket stand of 67 with Harrison Ward (40 off 21 deliveries, four fours, three sixes).
After their stand was broken, only two other Sussex batters made double figures as wickets tumbled on a regular basis.
The win sees Hampshire strengthen their grip on first place in the South Group, while Sussex remain rooted to the foot of the table.
- with AAP
