It was not a convincing performance from the Saints as they let a 12-0 lead slip away to trail 13-12 at the interval, before their pack began to dominate.
The home side scored four tries to two but poor goal-kicking and ill-discipline allowed Exeter to remain in the match right up until the final whistle.
Northampton made the better start with their forward drives troubling the visitors' defence and they had the first opportunity for points.
James Scaysbrook gave away a penalty 30 metres out but Ryan Lamb mis-hit his kick badly wide.
Exeter had their first chance of points in the 11th minute but Ignacio Mieres was also unsuccessful with a 30-metre penalty.
Some elementary errors from the Chiefs allowed Northampton to continue to dominate in terms of territory and following some incisive running from Dom Waldouck and Hartley, both Soane Tonga'uiha and Ken Pisi were hauled down narrowly short.
The Saints deservedly took the lead after 18 minutes when Ben Foden first fielded a clearance kick from Mieres, then evaded a couple of defenders to send Vasily Artemyev away.
The wing ran 30 metres to provide the scoring pass for Christian Day but Lamb again missed with an easy conversion attempt.
Exeter responded when their scrum decimated the home eight to gain a penalty which Mieres missed from almost straight in front, 40 metres out.
Almost immediately, Northampton made them pay with their second try. Lamb made a neat break to set in a motion a fluent movement which ended with Phil Dowson driving over from close range for the try which Lamb converted with the most difficult kick of the afternoon.
After 30 minutes, Exeter scored their first try when Josh Tatupu broke strongly down the flank to send Phil Dollman over with Mieres converting.
Lamb had a chance to extend Northampton's advantage but his 25-metre drop-kick was badly wide before the outside-half conceded a penalty with a high tackle.
That allowed Mieres to first kick that goal and quickly another as home number eight Gerrit-Jan van Velze was yellow-carded for a late challenge.
Exeter failed to take advantage of Van Velze's absence and the scored remained at 13-12 to the visitors after 50 minutes.
It was at that stage Saints introduced Stephen Myler as a replacement for the disappointing Lamb.
Mieres kicked his third penalty as Tatupu left the field, sitting up on a stretcher before Northampton lifted the tempo and regained the lead.
Tom Wood won a line-out on halfway, Ken Pisi tore down the right flank and when the ball recycled, replacement Burrell powered through some poor Exeter defence for a try on his home Premiership debut.
Northampton collected their bonus point when Hartley burst over from a line-out 10 metres out and Myler's conversion gave his side an eight-point lead with 15 minutes remaining.
Myler missed with a late penalty, allowing Exeter a slim chance.
Rhys Oakley was sin-binned for a foolish trip and Matt Jess crossed for a late try but it was too late to matter.
- Ignacio Mieres