Needing another 132 for victory with eight wickets in hand on the final day at Grace Road, Wells finished unbeaten on 90 as Sussex strolled home on 234-5.
Anchored by Wells' 190-ball innings, Sussex never looked in any trouble despite losing three wickets.
Sussex have now won back-to-back games, while Leicestershire are still winless.
Their successive victories have come under new captain Chris Nash after Luke Wright stepped down from the role at the start of the month.
Wells has been instrumental in both wins, scoring 155 in the innings victory over Worcestershire, before his match-winning knock in this game.
Sussex head coach Mark Davis told BBC Radio Sussex:
"In our other two wins this season we've been ahead of the game - but here we've come from behind to show loads of character, skill and guts. It was nice to see.
"A couple of individuals will take the headlines, but it was a proper team performance."
Luke Wells continued his superb batting form to guide Sussex to a five-wicket win over Leicestershire in Division Two of the County Championship.