Lincolnshire made one change, Luke Robinson replacing Jonny Tattersall.
On winning the toss Lincolnshire opted to bat first.
They quickly lost early wickets and at 28 for 3 needed to consolidate their innings.
Dan Freeman and Dominic Brown added 41 before Freeman was dismissed with the total on 69 for 4 after 22 overs.
Brown followed for 30 with the total 90 for 5. Four further wickets fell for only 29 runs and it was not until David Lucas joined Martin Weightman that some impetus was injected into the innings.
These two put on 40 with Weightman making 31 and Lucas 19.
A final total of 167 all out in the 48th over was unlikely to be enough to secure victory unless Lincolnshire could take early wickets.
With Norfolk at 59 for 3 and then 97 for 5 the match was evenly poised but a stand of 71 by the Norfolk captain, Ashley Watson (42 not out) and James Hale (25 not out) saw Norfolk to victory by five wickets with almost six overs to spare.
Lincolnshire next face Cumberland at Bracebridge Heath on Sunday (11am) when they will need to win to have any hope of qualifying for the quarter final stages of the competition.