New Zealand all-rounder Neesham was making his first appearance for the County having concluded his IPL commitments, and he got off to a flying start, lashing 75 from just 30 balls.
But he wasn't the Steelbacks’ top scorer as Australian ace Lynn stole the show with a stunning 106 from 66 deliveries, hitting 12 fours and three sixes.
Northants, who won the toss and opted to bat, racked up a whopping 227 for one from their 20 overs.
And Leicestershire, always under pressure in reply, were eventually limited to 185 for nine, with Neesham again having a say as he took two for 26 from his three overs.
Freddie Heldreich and Ben Sanderson bagged three wickets apiece as the Steelbacks secured an impressive 42-run success.
It was a second win from four games in the competition for Northants and they are now up to third in the embryonic North Group standings.