The match was reduced to 42 overs because of the stormy conditions that played havoc with so many fixtures across the region.
Barnsley were restricted to 118-6 with Nathan Swift hitting an unbeaten half-century and even though Duckworth Lewis increased the Doncaster target to 121, it looked a formality when openers James Stuart (47) and Duncan Heath (21) put on 70.
Daniel Waldron and Oliver Jackson started to chip away and there were only two wickets remaining when Luke Townsend and Curtis Free saw Town over the line.
Promoted Elsecar came within a whisker of causing the upset of the day against champions Wakefield Thornes who set a target of 207-8 mainly thanks to David Toft (42) and Matthew Jordan (43).
Matthew Higgins (47) gave Elsecar a solid start but their hopes soared during a 61-run sixth-wicket stand between Tanzeel Ali (38) and Charlie Kaye (43), only to be dashed when the last wicket fell five runs short of victory.
In the only other completed match, Jack Simmonite (55) and a quickfire unbeaten 41 from Adam Clarke helped Aston Hall to 196-8 but a third-wicket stand of 70 between Danial Rafiq (58) and Alex Fletcher (46) put Whitley Hall on the way to a four-wicket win.
In Yorkshire Premier North, Castleford’s 162, which included 47 from Scott Hopkinson, was revised to 94. Dunnington passed it, Adam Sutcliffe (37) and George Drury (36no) leading the way.
None of the Bradford Premier matches managed to finish but several players gave notice that they are already in form. Bradford & Bingley’s Jack Hartley snapped up 5-50 as Lightcliffe fell for 179; David Stiff took 6-37 as Hanging Heaton held Farsley at 131-8; Chris Brice (5-36) was among the wickets as New Farnley made 116-9 and Alex Lilley picked up 4-30 before Woodlands were halted by rain at 89-4.
Irfan Rafique (54) and Harry Warwick (63) shared a century stand in Townville’s 181-9 at Cleckheaton.
In the only Huddersfield Premiership match played, Shoukat Ali hit seven fours and three sixes and put on 73 with Jason Grosvenor (58no) as Moorlands overtook Armitage Bridge’s 175 with seven wickets in hand.
Aire Wharfe champions Otley raced to 243-1 through James Davies (89no) and Alex Atkinson, whose unbeaten century included 13 fours and five sixes, putting on 174. A declaration helped them dismiss Ilkley for 139.