Hain, the top run-scorer in last year's competition, helped his side reach their revised target of 159 in a game reduced to 30 overs.
He was aided by Will Porterfield's 63 in an unbroken first-wicket stand.
Northants struggled to 154 all out in their 30 overs, with Rob Keogh (64) the only one to make a real impression.
England international Ben Duckett was caught by Hain for seven off the bowling of Rikki Clarke (2-18), one of seven Northants batsmen to fail to get out of single figures as Oliver Hannon-Dalby (3-24) impressed.
Hain returned to the side having been left out of their previous County Championship game, on the day he received a reprimand from the ECB's Cricket Discipline Commission.
He was reported by the umpires during Warwickshire's Second XI Championship match against Worcestershire at Flagge Meadow last week, for a Level One breach of the code (bowling a fast short pitched ball and/or accidental high full pitched ball that results in the bowler being disallowed from bowling any further in that innings).
The penalty is a reprimand, which will remain on his record for two years. The accumulation of nine or more penalty points in any two-year period will result in an automatic suspension.
Hain scored more runs in the One-Day Cup last season than any other batsman (540 in 10 innings).
Northants coach David Ripley told BBC Radio Northampton:
"We just haven't come out of the blocks at all. We've been very well beaten. It's a pretty grim effort all round really.
"We had the sharp end of the conditions batting first, we had to stop start and couldn't get any partnerships together. But still our performance is way down from where we expect it to be.
"I expected our bowlers to be a handful and I was hopeful we could pull off something special but we never got started with the ball either."
Warwickshire batsman Sam Hain told BBC WM:
"I felt a bit of pressure coming into this competition so it's nice to get on the board here with a few runs.
"Especially after the start to my season in the red-ball game. I'm a little stumped to why it hasn't happened in the Championship.
"But I have no qualms about my white-ball game and now I have a block of games to work with now."
Sam Hain's unbeaten 89 helped holders Warwickshire race to a 10-wicket victory over Northants in a rain-affected One-Day Cup match.