Heath were set a revised target of 254 after Cuckfield’s innings was cut short by four overs.
And Jethro Menzies (72) and Max Barson (68) helped them get there with just one ball to go with Chris Blunt hitting the winning runs.
Ben Candfield took 4-44 for Cuckfield.
Earlier, Nipun Karunanayake (61) and Ollie Graham (71) starred for Cuckfield.
The other promoted side - Hastings and St Leonard’s Priory (200 all out) - also enjoyed victory, beating Middleton (147-9, revised target) by 15 runs.
Former Sussex star Harry Finch top-scored for Hastings with 62 while Sean Heather hit 53 for Middleton.
August Cup winners Eastbourne (163-8) enjoyed a tight two-wicket victory against reigning Premier Division champions Roffey (160-6). Rohit Jagota (57) top-scored for Roffey in their 235 over innings.
Harnoop Kalsi (4-14) bowled well for Roffey as Eastbourne wickets fell regularly but Ben Twine (37) and Joe Pocklington (45) helped the Saffrons side get over the line.
Three Bridges are yet to finish a game after their trip to Brighton and Hove was abandoned.
Preston Nomads (206-8) beat East Grinstead (133 all out, revised target) by 61 runs.
In Division 2, Bognor (159-1) recorded a very convincing nine-wicket win against Horsham (158-8 from 40 overs).
Ryan Maskell (56) and Nicholas Ballamy (75 not out) helped see Bognor home.
Chichester Priory Park (166-4) also wicked up a win as they were set a revised DLS target after a rain hit match at Lindfield (114-3).
Reduced to 43 overs a side initially, Lidnfield’s innings was cut short to 28 overs at 114-3. Chichester also received 28 overs and needed to get 163 to win and cruised to the total for the loss of four wickets. For Chi, Mike Smith hit a 19-ball 38 including four sixes.
Burgess Hill (190-8) enjoyed a 90-run win over Billingshurst (100 all out). Wajid Shah (50) and Lewis Mamoany (58) were the stars with the ball for Hill, while Jas Bassan took 4-15 with the ball.
The other two matches in division two were abandoned.
Division 3 West: Broadwater (154-2) beat Henfield (153 all out) by eight wickets with Ben Challen taking 4-25.
Findon (186 all out) lost to Slinfold (190-5) by four runs with Greg Iago hitting 63 not out for Slinfold.
West Chiltington & Thakeham (309-6) beat Pagham by (159-6) by 150 runs with Hugo Gillespie (73) and Ben Lucking (50) top scoring. Theo Benyon Ayres hit 51 not out for Pagham.
Worthing (185 all out) beat Littlehampton, Clapham & Patching (108 all out) by 77 runs with Girgio Rigali taking 5-25.
In Division 3 East, Glynde & Beddingham (219-4) beat Brighton & Hove 2nd XI (217 all out) by six wickets. Fergus Guppy hit 86 for Brighton while Archie Burrows (54) and an unbeaten 94 from Charlie Hobden saw Glynde home.
Ifield (134 all out) lost to local rivals Crawley Eagles (159-8) by 25 runs. Rehan Hasan hit 53 for Eagles before Atif Ali took 4-26. Skipper Jack Groves hit 74 for Ifield.
Other notable performances:
Division 4 West: Gary Mockford took 5-15 for Chippingdale as they bowled Felbridge and Sunnyside out for 51.
Division 4 East: Ben Smit took 6-34 KKeymer and Hassocks in their win over Isfield. Jake Lingard took 5-19 fior East Grinstead 2nd XI against Mayfield 2nd XI.
Division 7 West: Priya Chameera Dondeeru took 5-14 food Slinfold 2nd XI against Crawley Eagles 3rd XI.
Division 8 West: Peter Boyce hit 111 for Henfield 2nd XI against Barns Green 2nd XI.
Division 8 Central: Chris Hinton smashed 112 from 63 balls and Ben Heller took 5-17 for Felbridge & Sunnyside 2nd XI
Division 9 Central: Leroy Dekker took 6-36 for Three Bridges 3rd XI and Rob Littlejohns 5-9 for Scaynes Hill 2nd XI.
Division 9 East: Henry Hamon took 5-23 for St Peter’s 3rd XI.
Division 10 East: Sam Aldis took 5-18 for Nutley. Two one -wicket wins - Nutley and Robertsbridge.
Division 11 East (South): Hollie Young took 7-17 for Sidley against Battle 2nd XI.
Division 12 West (North): Andy Parsons ht 140 for Slinfold 3rd XI.