Neesham hit an eye-catching 75 from 30 balls as he teamed up with Australian opener Chris Lynn, who lashed a stunning 106 from 66 deliveries, to help the Steelbacks make a massive 227 for one.
The Foxes never looked like chasing it down, and there was more delight for Neesham as he claimed figures of two for 26 from three overs to help limit Leicestershire to 185 for nine.
“When you come into a new team, you want to prove your worth in the first couple of games," Neesham said following Wednesday evening’s win.
"I've been watching from afar and Lynny’s (Chris Lynn) done that already so far in the tournament.
"So, it's just a real pleasure to come out and first of all, get the win and spend a bit of time with one of my mates out in the middle and have a good time."
Neesham's arrival had been delayed by his IPL commitments and a visa issue.
“I actually mucked my visa up a little bit," he explained. "So I had to fly to Paris after I got here.
"I woke up this morning and had an omelette du fromage in Paris and hopped on a plane and landed at about midday.
"I got a bit of sleep and then came to the ground.
"I was thinking if I could get through three or four overs and have no injuries and have a bit of a hit, I’d be pretty happy.
"So to have a result like that, and obviously to witness Lynny bat the way he did from up close is a 10 out of 10 evening."
Neesham clearly loved batting alongside Lynn, who has also made a big initial impression at Northants.
“Lynny is fantastic," Neesham said. "He’s one of the best ball strikers in the world. Probably one of the best ball strikers in the history of T20.
"We've spent a lot of time together at different teams, and we enjoy each other's company a lot.
"It's just fun being out there with him. We know each other's games so well, so we can recommend things to each other and keep each other in line.
"We probably worked together quite well in that partnership.
"Hopefully, he can continue to keep scoring runs.
"I quite like batting three as well.
“Lynny’s a strong man. If you see him up close, he's got some big arms on him and he's a very stocky, well-built guy.
"With the big sixes that are being hit in T20 cricket, everyone talks about the bats but no-one talks about the extra work guys are doing in the gym these days to be more powerful and more explosive. He's probably testament to that.”