Kent 222 (Crawley 91, Tremain 5-44, White 4-57) and 227 for 3 (Compton 114*, Leaning 67*) beat Northamptonshire 117 (Vasconcelos 39, Quinn 4-25, Stewart 3-6) and 331 (Keogh 116, Berg 56, Azad 51, Evison 4-62) by seven wickets
The visitors had looked heavy underdogs at the end of day two, when they were still 58 behind with eight second innings wicket remaining, but after batting through Saturday they began the final morning on 300 for 7, 195 in the lead.
Although Kent polished off the tail relatively quickly, the additional 31 runs Northamptonshire stockpiled ensured a jittery couple of hours for the home fans, until Compton and Leaning bedded in.
Berg reached fifty from the second ball of the morning, with a single off Michael Hogan, but the bowler took revenge in his next over, sending Berg's middle stump flying.
Evision wrapped up the innings with two quick lbws, removing Jack White for 7, then getting Chris Tremain for a duck.
It was a target that looked some way off when Zak Crawley had his stumps splayed by Chris Tremain, having made just 3. Daniel Bell-Drummond responded with a rapid 32 from 31 balls but Berg then ripped out his off stump and when Tremain then bowled Joe Denly for 5, Kent still needed 167.
Compton's initial caution faded after he reached fifty by pushing White through the covers for four. He was nearly out on 53 when he edged Keogh just past Ricardo Vasconcelos at thirst slip and he responded with a reverse-swept four.
Leaning's diligence meant Kent were on a far healthier-looking 158 for 3 at tea and he subsequently nudged a single off White to bring up his half-century.
On Easter Sunday last year Compton had scored 115 in a heroic but doomed rearguard action against Lancashire. This time round he swept Keogh to fine leg for four to reach three figures and the victory was confirmed shortly afterwards when Leaning scrambled a single off the same bowler. Kent take 19 points, Northamptonshire three.