Darren Stevens’ 14th first class five-wicket haul ripped out the Sussex top order before Mitch Claydon, Matt Coles and Wayne Parnell helped wrap up a 226-run win at Hove.
Kent moved to within a point of the top of the Specsavers County Championship table with 22 points after an overnight declaration left the hosts facing a target of 427 for an unlikely win.
Sussex counter-attacked in the early overs bowled by Parnell and Coles with the opening stand of 59 before the veteran Stevens struck five times before lunch to round off a wonderful all-round display.
The openers looked comfortable until Finch played a loose shot to a wide one before Nash nicked behind to Rouse to start a collapse.
Nash was the first of five wickets to fall for 17 runs in eight overs. Kolpak signing Stiaan van Zyl was next to fall, edging Stevens to slip before the veteran all-rounder trapped Laurie Evans LBW for 0.
Mitch Claydon grabbed Kent’s sixth when Luke Wright was adjudged leg-before.
After successive half-centuries, Stevens and Claydon stifled the scoring and left the hosts reeling on 93 for six at the interval.
Ben Brown, Sussex’s standout batsman in the match, was stranded on 69 not out as the last four wickets fell to catches.
Coles returned at the Sea end and tempted former South Africa all-rounder David Wiese into hooking to Daniel Bell-Drummond at deep square leg.
His compatriot Vernon Philander soon followed - picking out Joe Denly at mid-wicket before Jofra Archer nicked him to sub fielder Adam Ball at slip.
Kent’s overseas import Wayne Parnell wrapped up the win as he tempted Ajmal Shahzad to square cut to Denly at point.
Kent captain Sam Northeast said: “We played some really good cricket last year and now we’ve gone up another notch.
“Darren bowled an unbelievable spell. He just keeps on doing it. He’s got fantastic skill with the ball. He set it all up. He’s got a fantastic wrist. He can nip it both ways holding the seam up. He bowled hardly any bad balls. He’s not the quickest but he’s got unbelievable skill.
"Mitch Claydon backed it up and then Matt Coles bowled a really good spell at the end. We didn’t have a very good half-hour with the new ball but after that we were relentless and put in a great spell.
“Our bowling attack has got some really good variety. Claydon slams the wicket, the same with Coles, and then we have the left-arm variety with Wayne Parnell and the spin of James Tredwell. It’s a nice attack for me to call on.”