Spinners Keogh and Simon Kerrigan both claimed five wickets as the home side collapsed from 64 without loss to 161 all out, having been set 365 to win in around 73 overs.
And it was less than 10 minutes before the end of play that Keogh struck, ending last man Matthew Quinn's stubborn effort to dismiss him for a 37-ball duck.
The wicket sparked ecstatic scenes among the Northants players, as they claimed a first County Championship top flight win since they beat Surrey in July, 2004.
It is also a win that sees the County move up to fifth in the table, and 31 points above the bottom two with six matches remaining.
Keogh, who ended with five for 31 from 21.5 overs, said: "We’re tired, but very, very happy.
"It’s not very often that you get four-day wins that go into the last five minutes or so left on day four.
"I think everyone’s drained and gave it their all so to get it over the line that late … what a feeling!
Northants had continued their second innings on Thursday morning before declaring on 296 for eight with Ryan Rickelton making 133.
And Keogh admitted it was a tough decision on what sort of a target to set Kent.
“With Baz Ball knocking around the corner you never know what sort of a total you need," admitted the off-spinning all-rounder, referring to the England Test team's recent run-chasing exploits.
"The pitch was fairly flat at times, but we just thought if we can bat them out of the game, try and make it impossible for them to chase, or if not impossible then for them to be attacking and give us chances that it would give us chances.
"It turns out they didn’t really have a go at the carrot and it just meant we could go ultra-attacking around the bat.
“I think the pressure told on them a bit, if you look at Jack White up front I think he bowled eight overs for eight runs and he could have got Zak Crawley out at any point.
"It meant the pressure was too much for them, Simon started it getting those quick wickets and it just meant that we could build.”
Left-arm spinner Kerrigan started the Kent collapse as he claimed the crucial first three wickets - including those of England man Zak Crawley and run machine Ben Compton - and ended with five for 43 from his 28 overs.
And he felt the dismissal of Crawley was a key moment.
“If you look at the Kent batting line-up, it’s full of either Test cricketers, ex-Test cricketers or lads who’ve got multiple centuries to their name and they bat quite deep as well," he said.
"But thankfully the pitch did wear quite a lot, especially the footholes.
"It was spinning out of that which obviously helped massively have Rob (Keogh) there as well to spin it into the right handers.
“I felt once we got Crawley out we got our tails up a little bit. It was just a case of keep trying to do the right things and trust there’d be a bit of pressure.”
Northants now have a few days off before they entertain third-placed Lancashire at the County Ground, with the Championship clash starting next Tuesday.