They kicked off their cup trail at Grimsby Borough Development on Saturday and found themselves 2-0 up at half-time.
Grimsby fought back after the break, but the visitors came away with victory to take into this Saturday’s home leg at London Road.
Aron Burton and Jed Stainton were on target, while Ellis Barker chipped in with his 24th goal of the season.
Alford Town secured their place in the second round of the cup after North Thoresby conceded both legs, and begin their two-legged tie at Susie Star on Saturday.
Louth Old Boys Reds booked their place in the knockout stages of the Tom Sandys Trophy after beating their clubmates on Saturday.
Michael Brown hit a double and Jordan Tebbs found the net to give Reds a 3-2 win at Louth Old Boys Vets.
The Reds can now finish no lower than second in Group A and will top the group with victory against leaders North Somercotes United Reserves.
Alford Town Reserves, meanwhile, ended their league campaign in Division Three of the East Lincs Combination with a 2-0 home defeat to AFC Grainthorpe.
Goals from Cavan Campbell and Adam Manson consigned Alford to a second-bottom finish.