The Cleckheaton innings was halted after just 4.1 overs as a torrential downpour held up proceedings for over two and a half hours.
When the match resumed, shortly before 4pm, Hartshead's bowlers took full advantage of the conditions to help reduce the Moorenders to 87-7 after 29 overs.
A massive cup shock looked on the cards until Craig Thornton and Rhys Jowett came together and produced a patient, well crafted 89-run partnership for the eighth wicket which helped Cleckheaton reach 183-8.
In the first over after the rain delay, Joel Pollard trapped Louis Aspeling (3) lbw and celebrated with a forward roll.
Four overs later Ian Wood cut Pollard for a lovely four but the very next ball, he got one to pop just short of a length and Wood got a top edge and was caught by Danny Squire at gully.
John Wood (13) was bowled by Squire before Mally Nicholson (0) was caught behind by Chris Mould off the bowling of Moor captain Simon North and when the Bradford League's leading run scorer, Mohammad Zaman, was bowled by North for 16, Moor had real hope.
Ian Nicholson (1) drove Squire and was out to a well-judged catch by Matthew Evans at mid-off as Cleckheaton slipped to 64-6.
Thornton got off the mark with a fine straight six off Squire but then Neil Nicholson (10) was bowled by Martin Ellis as Cleckheaton slipped to 87-7.
You got the feeling that if Moor could have taken one more wicket, Cleckheaton would have been in serious trouble.
Thornton survived one scare when he was dropped, Pollard, diving full length at short fine leg, with the score on 107.
Thornton hit two sixes and two fours in a 66-ball half-century and ended 56 not out from 76 balls, while Jowett was eventually bowled by the returning Pollard in the 49th over.
It proved a match-winning partnership as Cleckheaton closed on 183-8 and it was always going to prove a challenging total for Hartshead to chase.
Moor were soon in trouble as Iain Wardlaw and Craig Wiseman bowled devastating opening spells.
Wardlaw spent the winter playing cricket in Australia and he looks to have added an extra yard of pace and at times was simply too quick for the Moor batsmen to play.
In his first over, Wardlaw got North to edge one and John Wood snaffled a reflex catch at slip, while Graham Hilton edged Wiseman behind to Mally Nicholson.
Evans clattered Wardlaw to the boundary on a couple of occasions but each time Wardlaw replied by beating the batsman for pace.
Wardlaw dug one in short and Evans hooked it away for two before the paceman bowled a superb swinging yorker to bowl Evans for 13.
Ellis and Field steadied the ship and produced a 37 run stand before Field (8) was run out by a fine pick up and throw by Jowett.
Ellis (22) was caught behind by Mally Nicholson off Ian Wood before the same combination accounted for Danny Squire.
Zaman picked up the wickets of Nick Ward and Matthew Waudby before the returning Wiseman bowled Trent Otto.
It was fitting that Thornton wrapped up the 79-run win when he bowled Pollard for 14 at set up a quarter-final tie at Farsley or Bowling Old Lane on July 13.