Church pro Saeed Anwar ripped through the East Lancs order on Saturday with 7-29.
They skittled the Blackburn side out for 76 but then wickets fell at regular intervals for Church in a low-scoring game.
But 19 by bowler Andy Bentley helped them pass the target with three overs to spare.
Enfield pro Werner Coetsee celebrated a half-ton – smashing 55 from 65 balls – in their 155-5 against Rawtenstall.
Opener Andy Payne’s 62 looked like it would secure a win for the Valley side but no one could stay with him for long as pro Coetsee celebrated a good day all-round with five wickets as Enfield won by 40 runs.
Rishton couldn’t conjure up their first win of the campaign against fellow early-season strugglers Burnley.
Despite the top four all getting in their 20s the middle and lower order couldn’t get out of single figures as they collapsed from 128-5 to finish 136-9.
And Burnley passed the target with a loss of just three wickets.
Accrington’s clash at Bacup was the first game of the new season to fall victim to the weather on Sunday.