Jayasuriya took three for 39 in his nine overs to limit the Warriors to 241 for nine and lift the Dolphins into fifth place on the log.
Captain Davey Jacobs hammered 73 off 71 balls to take the Warriors close as they chased 251 for victory, but the visitors were always going to pay for the lack of a substantial innings from their top order.
One of the basic rules of successful limited-overs cricket is that one of the top five should bat through the innings, but Michael Smith (54), Jon-Jon Smuts (24), Arno Jacobs (25) and Zander de Bruyn (20) all made starts but failed to press on.
The Dolphins made the match more dramatic than needed to be by dropping a couple of catches in the closing overs, but Yusuf Abdullah then bowled a crucial penultimate over that cost just six runs.
Jayasuriya was called into service to bowl the 43rd and 45th overs, and, if he hadn't have bowled Davey Jacobs with the fourth ball of the final over, the outcome may have been different.
All-rounder Jon Kent once again made a telling impact, conceding just 22 runs in his five overs and having Smith caught behind.
A day/night campaign that was rapidly going down the toilet for the Dolphins has now been revived thanks in the main to a top-class partnership between Imraan Khan and Ahmed Amla.
The pair added 155 for the second wicket to provide the foundation of the Dolphins total of 250 for seven.
After Jayasuriya (13) had committed the sin of following three fours in four balls by being caught at third man off Juan Theron, Khan and Amla made batting look easy as they lifted the Dolphins to 171 for one in the 33rd over.
The captain and the in-form opener ensured they took few risks early on and it needed the introduction of a bowler who has bowled just two overs in the competition this season to finally send Amla packing.
The skipper tried to belt slow left-armer Smuts out the ground but holed out to Smith on the midwicket boundary.
Amla had nevertheless caned five fours and a six in a punishing 64 off 86 balls.
Khan is the most successful opener in the competition so far in this campaign and he produced another top-class innings of 86 off 118 balls before being bowled by talented left-arm paceman Wayne Parnell.
Useful run-a-ball contributions from Kent (27) and Pierre de Bruyn (21) then ensured the Dolphins' run-rate stayed above 5.5 runs an over.
De Bruyn fell off the last ball of the innings when he hooked Parnell high and handsome to the midwicket boundary, but Lyall Meyer produced an exceptionally well-judged leap and when he returned to the turf he had claimed a superb catch.
Parnell finished with four for 56, while De Bruyn's nine overs for 33 runs was a top-class effort.
Smuts finished with one for 31 in six overs, while Theron, Meyer and Parnell would all go for more than six runs an over.
If the Dolphins had managed to restrict the Warriors to 200 or less, they would have earned a bonus point that would have taken them to 10 log points and pushed them into fourth place ahead of the Eastern Cape team.
Nashua Dolphins - Imraan Khan, Sanath Jayasuriya, Ahmed Amla, HD Ackerman, Jon Kent, Daryn Smit, Pierre de Bruyn, Johann Louw, Quinton Friend, Yusuf Abdullah, Ugasen Govender.Chevrolet Warriors - Michael Smith, Jon-Jon Smuts, Athenkosi Dyili, Arno Jacobs, Zander de Bruyn, Davey Jacobs, Johan Botha, Craig Thyssen, Wayne Parnell, Lyall Meyer, Juan Theron.