Daniel Higgins put the Scottish League Two champions in front at Balmoral Stadium with an expertly-taken free-kick in the 17th minute.
But Hibs were transformed after the break and Nisbet had a hand in substitute Jamie Gullan's equaliser four minutes after the restart.
The former Dunfermline striker then volleyed home on the hour mark to seal a 2-1 victory for the visitors.
Hibs came close early on when Christian Doidge headed against the bar from Martin Boyle's cross.
But Paul Hartley's side were imposing themselves well on the game and Higgins curled a free-kick over the wall and past Dillon Barnes in the Hibs goal.
Hibs had much of their defensive unit unavailable, with Ofir Marciano, Ryan Porteous, Paul McGinn and Alex Gogic on international duty and left-back Josh Doig self-isolating after Scotland Under-19s coach Billy Stark's positive Covid-19 test.
But they still had an experienced back line, with the likes of David Gray, Darren McGregor and Lewis Stevenson featuring.
So head coach Jack Ross would have been concerned to see the defence caught out by a quick free-kick that set up Mitch Megginson. The striker stabbed the ball towards goal and Barnes gathered at the second attempt.
Stevie Mallan set up Doidge with a ball over the top, but Stuart McKenzie stood up well to the striker's powerful effort.
Ross brought on Gullan for Drey Wright at the interval and Hibs were quickly level. Nisbet ran at the heart of the Cove defence and laid it off to Boyle, whose square pass was stroked into the net by Gullan.
Nisbet's seventh goal of the season was a real touch of class. The forward guided a looping volley into the top corner after running on to Melker Hallberg's ball into the box.
Hibs looked in control but needed Barnes to be lively to tip over Motherwell loan forward Jamie Semple's brilliant 25-yard volley with a minute remaining.