The victory means Notts will finish second or third in Championship Division One, depending on the result in the Warwickshire v Somerset match.
Set a target of 174 to win, they lost four wickets after resuming on 42-1, but Slater guided them home on 174-5.
Having failed to win any red-ball games in 2019 or 2020, Notts have seven victories this season and their total of 73 points in Division One puts them just 0.5 behind Lancashire, who beat Hampshire on Thursday.
Ben Duckett beat Slater – who was dropped on 19 – to 50 at the start of play, reaching his half-century with a paddle sweep for four off Dom Bess.
But a loose drive away from his body on 54 saw him play on to Jordan Thompson and he swished his bat over the bails in annoyance before walking off.
Joe Clarke (10) skied a catch to mid-off to give Bess a wicket and Ben Coad (3-58) bowled Steven Mullaney for nine as Notts reached lunch on 149-4.
Tom Moores was caught at mid-wicket off Coad following the resumption, but Slater was unmoved and able to watch from the other end as Liam Patterson-White clipped Bess away for the winning boundary.
Nottinghamshire: 296 (Clarke 109, Evison 58) and 174/5 (Slater 79*, Duckett 54)
Yorkshire: 73 (Evison 4-13, Fletcher 3/31) and 396 (Lyth 153, Paterson 3/39, Hutton 3/101) (BBC)