Dan Barratt hit 88 and Collis King 34 in Dunnington's 198 all out in 38.3 overs, with Mark Askham taking 3-34 for the hosts.
Andy Tute spearheaded the reply with 101 and Dave Sykes made 27 as Acomb squeezed home with two wickets and two balls to spare.
Easingwold, last season's beaten finalists are safely through to a second round clash with either Studley Royal or Folkton & Flixton after a 69-run win over Sewerby.
Easingwold compiled an imposing 265-9 from 40 overs, with Paul Skilbeck top-scoring on 93 while Steve Piercy hit 46 and Mike Mierzwinski 38.
The visitors closed on 196-6.
Clifton Alliance progressed with a round one victory at Sheriff Hutton Bridge.
Hosts Bridge batted first and were bowled out for 105 in 38 overs, with Dave Phillips hitting 25, a score matched by Alex Ibbotson. Alliance's Green claimed 3-41.
Adam Slater top-scored with 48 as the visitors booked a second round trip to Fenner with a four-wicket triumph in 33.1 overs.
Ben Harrison took 3-15 for Sheriff Hutton Bridge.
Stamford Bridge secured a tie at Woodhouse Grange in round two as they won by five wickets at Beverley Town.
Dave Chaplin took 4-15 and Dan Davies 3-21 as Beverley were dismissed for 105 in 33.2 overs, with Stamford easing home in 26.4 overs.
Grange got the better of Sessay by 32 runs at Sutton-on-Derwent, with all-rounder Tom Young playing a starring role.
The hosts totalled 201-6 from their 40 overs, with Young leading the way with an unbeaten 106, backed by Nick Hadfield's 29 and Chris Suddaby's 26 not out.
Young then took 3-27 as Sessay were bowled out for 169 in the 39th over.
Hunters Premier Cup, round 2 draw: Fenner v Clifton Alliance, Acomb v York II, Woodhouse Grange v Stamford Bridge, Folkton & Flixton or Studley Royal v Easingwold. Ties to be played on Sunday, June 29.