The Cherry and Whites struggled for the opening 60 minutes of the match, but when Louis Rees-Zammit scored 17 minutes from time, it sparked the home side into life.
Gloucester then took control thanks to some hugely dominant lineout drives, which led to a penalty try and a yellow card for Manny Iyogun.
And Skivington's side finished the job with a score from Alex Craig, again from an unstoppable maul.
It left Saints deflated as they suffered their fourth successive loss in the Gallagher Premiership, while Gloucester boosted their bid for a Gallagher Premiership play-off place.
"I thought the half-backs managed the game really well," said Skivington, who saw his side secure a bonus-point success.
"There were a few switch-off moments and Saints pulled some real class out of nowhere.
"It was fast and furious, but it's just nice to get the win.
"We were 13 points down and felt like we were stuttering a little bit, but the boys just stayed on task and eventually ground it out so that's very pleasing.
"I'm not too worried where we sit in the table, I just want us to get better and better.
"And games like this where we're not quite in our rhythm, if we can grind those out then we're in a much better spot than we were last year."