Phil Dowson added Northampton's second in 28th minute, after the Saints capitalised on another Exeter mistake.
The Chiefs responded as Josh Tatupu beat Alexander Artemeyev for pace, sending Phil Dollman under the posts for Exeter's first try.
Two penalties from boot of Mieres sent Exeter into half time with a 13-12 lead.
Exeter's play was littered with mistakes and they were fortunate to extend their lead to 16-12 early in the second half.
Luther Burrell then powered through a small hole in the Exeter defence to regain the lead for Northampton, setting up a one point game at 17-16.
Dylan Hartley bundled over from a well-worked Saints line-out move shortly after and looked to have secured the win for the Saints at 24-16.
Matt Jess scored a try late on to rescue a bonus point for the Chiefs, but the Saints managed to hold on for the win.