Sleaford reached 50-2 thanks to Matthew Mountain (31) and Charlie Roberts (16), but then lost three wickets for just two runs as opening bowler Jake Harrison (4 for 36) took charge.
The visitors looked set to be bowled out for double figures at 70-8.
Angus Youles (32) spared Sleaford’s blushes with a 43-run ninth wicket stand with Mason Mansfield (13 not out) before they were eventually all out for 125.
Grimsby’s reply started badly as skipper Shaun Morris bowled opener Joe Farmery and Nick Goacher has skipper Neal Snell caught behind by Oliver Bean to leave the hosts faltering on 12-2.
But the in-form Steve Crossley added 42 with Geoff Middleton and then another 51 with Hayden Tice to take Grimsby to 93-3.
Sleaford hauled themselves back into the game once more with three wickets for just six runs, as Morris (3 for 19) removed Hayden Tice and Jack Nilsson, while Youles finally dispatched Crossley for 54.
But Jack Harrison and Harrison Tice mounted a rearguard action to take their side to 124 before Tom Thompson hit the winning boundary off his first ball to win the match by three wickets.
Sleaford: M. Mountain 31, C. Roberts 16, K. Colyn 12, A. Hibberd 1, O. Bean 0, S. Morris 2, A. Sreedharan 1, A. Youles 32, C. Hall 0, M. Mansfield 13, N. Goacher 5*, Extras 12. Total: 125.
Bowling: S. Morris 9-4-19-3; C. Hall 5-0-15-0; N. Goacher 9-4-28-1; M. Mansfield 4-0-15-0; A. Sreedharan 8.4-2-27-2; A. Youles 7-1-21-1.