The visitors moved on from their overnight 182-7 to 313 all out, giving them a first-innings lead of 188 runs.
Ollie Robinson (81) and Ben Brown (71) put on 133 for the eighth wicket.
Having been bowled out for 135 on day one, Leicestershire fared little better as paceman Jofra Archer (3-31) added to his four first-innings wickets.
David Wiese (3-20) helped Sussex to 22 points from the match - enough to lift them above Worcestershire and up to third in the Division Two table.
Leicestershire captain Mark Cosgrove:
"We didn't play well, and it was pretty disappointing. To put in a performance like that on the back of the defeat by Essex was poor.
"I think the effort and respect we should have showed wasn't there. We were lucky to be in the game this morning, thanks to Neil Dexter's efforts on day one, but no-one helped him out.
"I don't think there's been a reaction to the news [director of elite performance] Andrew McDonald is leaving, definitely not.
"This is professional sport, people move on, so I'd be pretty disappointed if I heard any of the boys talking about that. There are no excuses."
Sussex captain Ben Brown:
"It takes a lot of character to come back from losing as comprehensively as we did against Kent last week, but we have to understand we're a young side and there are going to be ups and downs.
"We had a long honest chat about moving forward, and this is a step in that direction.
"I felt there was a huge amount in the pitch, it wasn't a great one to be perfectly honest, and Ollie and I looked to counter-attack this morning. As we went on it got a little bit easier, as it always does, and we took it to the opposition, and a lead of 178 was always going to be decisive: to get over 300 on a 160 pitch was a fantastic effort.
"Jofra Archer is a great talent, he'll have good and bad spells, but he bowled some frightening deliveries, cleaning guys up with balls going like off-spinners at 85mph."
Sussex completed victory over Leicestershire by an innings and 59 runs inside two days by skittling them for just 119 at Grace Road.