Lincolnshire started solidly after being put in to bat, withJoseph Kendall (42) and Tom Keast 33 putting on an opening stand of 71.
Ben Coddington (51), Ben Wright (37) and Sam Johnson (41) all added useful contributuons.
But Suffolk took control of the lower order with Daniel Freeman, Nic Keast, Mark Footitt and Danny Cliffe failing to make double figures as Lincs posted 249 from their 50 overs.
Footitt dismissed both openers, with Danny Cliffe claiming two more to leave the hosts rocking on 129/6.
But a brilliant seventh wicket stand between Joe Gatting (81) and Alex Oxley (46) saw Suffolf home with six balls to spare.