Rinke was the all-round star for the hosts, the former Zimbabwe international snapping up three wickets for three runs in his four overs to help mop up the Harrogate tail.
Opening bowlers James Wainman and Jack Holt impressed with figures of 2-39 and 3-24 respectively as Harrogate wickets fell at regular intervals, the visitors finally all out for 124 in 35 overs.
Harry Stothard’s 42 was the only innings of note as the visitors struggled, openers Henry Thompson and Ben Kempley the only other batsmen to make it to double figures with 14 and 15 respectively.
Prince Bedi and Linden Gray also snapped up a wicket apiece for the home side.
Scarborough made a shaky start as openers Oliver Stephenson (5) and Duncan Brown (4) fell cheaply as they slipped to 18-2.
South African batsman Breidyn Schaper once again showed his quality with the bat, his 45 not out settling the nerves for Scarborough as he put on 45 for the third wicket with Ben Elvidge (18).
Wainman swiftly departed as Scarborough stuttered, but Rinke’s 26 from just 16 balls, including four fours and a six, helped Schaper steer the hosts to victory with five wickets to spare.
Ishan Abeysekara was the pick of the away team’s bowling attack with 3-46.
Scarborough head to York this Saturday, 12noon start.
Scarborough 2nds earned a winning draw on the road at Hull Zingari 2nds in York League Division One East.
David Snowball led the way for the visitors with a superb 92 as they posted 261-9, Ed Hopper adding 34 and brother Charlie Hopper 27. Simon Powderly took 3-59 for the hosts.
Zingari never got going in reply as they ended up hanging on for a draw with 143-8, Sam Carver taking 4-33 and Ryan Hargreaves bagging two wickets.
The 2nds are in derby action at home to title-chasing Bridlington on Saturday.
Premier Division high-flyers Folkton & Flixton, whose game at Pickering was washed out last weekend, host Welton on Saturday.