The stand-in skipper admitted he had stern words for his troops as the wheels came off midway through the Ageas Bowl clash.
But after a dramatic finale, in which Kent were set a victory target of 153 in 17 overs, seamer Abbott saluted his men.
Despite Joe Denly smashing 29 not out from 22 balls, the visitors gave up and shook hands when they slumped to 57 for three.
Abbott said: ‘It was pretty exciting towards the end.
‘Talking to a couple of spectators on the boundary, they were thoroughly enjoying it.
‘We had a few stern words earlier in the match where I didn’t think the boys had pitched up.
‘We were slightly off the boil.
‘We were in a great position after day one and we didn’t come with enough intensity on day two.
‘I felt we sat back and almost lost a lot of momentum.
‘I haven’t done a lot of captaincy in the past, maybe not since school days.
‘It was challenging especially when I was bowling.
‘But we came off a tough time last week and whenever I asked the guys to come back for another over they were incredible.’
Kent bowled fourth-placed Hampshire out for 298 to set up the chase late on day four.
The visitors’ first innings of 555 had put them in the driving seat after the hosts had posted 409 for nine declared.