George Scott (77) and Miles Hammond’s (67) half-centuries had given the visitors hope of chasing down 368 and a first victory of the season.
But the second new ball claimed four wickets - including two for Mohammad Abbas, taking his match haul to nine - to dent their dream. Fellow seamers Keith Barker and Kyle Abbott picked up one wicket each.
Abbas had taken six in the first innings when Gloucestershire, replying to Hampshire’s 342, were all out for 179.
Benkenstein said: ‘I think the same pattern has been happening most games. The most obvious thing to point to is first innings runs but at the same time we need to be realistic, we are up against a really strong team.
‘There was a lot of good bowling from them, it wasn’t just a case of bad batting, and injuries aren’t helping.
‘Today the guys hung in and stuck in there but a match-winning performance would have been one of them getting 140 or so. We were just a bit short on that and it has been tough.
“It was a brilliant knock but it is disappointing how George Scott got out. I like the way he was being positive but we needed to have wickets in hand before that second new ball as they are a really good new ball bowling team.
‘I haven’t been able to get the same bowling attack out twice. April and May is about your bowlers. If you have a strong bowling attack you don’t have to chase as many runs.’
Hampshire captain James Vince, whose side are eight points adrift of leaders Surrey, said: ‘The pitch at different stages of the game was hard to bat on and then when the ball got soft it was hard for the bowlers.
‘There was no real pattern to it but it happened at different times.
‘It was a hard-fought win. We got ahead in the game a couple of times but let them back in, but on the whole it was a fantastic win.
‘There was definitely more in the pitch with the new ball. While we only got one wicket in the morning we controlled the rate, as they were still in the game runs wise.
‘The way we bowled with the new ball and took those wickets took the game away from them, then we sustained the pressure.’
Hampshire’s next game is against Somerset at Taunton, starting on Thursday, May 19.