It means the final place in the end of season play-offs will be decided on the last day of the regular season - Saturday, May 5 - when Saints host Worcester Warriors and Exeter are away to Saracens.
Exeter also crossed the line twice, scoring first and last through Gonzalo Camacho and Richard Baxter, with five points from the boot of Ignacio Mieres.
Exeter were led out by number eight Baxter on his 350th league appearance for the Devon side, with Phil Dolman switched to full-back and Jason Shoemark coming in at centre.
Northampton made two changes from the defeat to Leicester which cost them a top-four place last week, with Myler replacing Ryan Lamb at fly-half and Mark Sorenson in for Samu Manoa in the second row.
The Saints made a strong start and the visitors got the first points on the board after five minutes with fly-half Myler landing a 23-metre penalty.
But the Chiefs hit back immediately with scrum-half Haydn Thomas taking a quick tap penalty and finding Camacho.
The winger stretched for the line but referee Andrew Small referred the decision to TMO Geoff Warren, who awarded the Argentina international his second try of the season. Countryman Mieres added the extras.
Saints again applied pressure on the Chiefs half but this was short lived and the home side were quickly back up the other end to batter the visitors line before Mieres added a simple penalty midway through the half.
Myler and Mieres were both wide of the target with penalty attempts before the Chiefs had a try turned down from the TMO after prop Ben Moon had knocked on over the line.
Exeter went in to the break 10-3 ahead.
The Saints started the second half strongly with some good passing amongst the backs helping Ashton over in the corner. Myler was again off target with the conversion, as was Mieres with a 40-metre penalty attempt.
The Chiefs struggled with their own lineouts and, with the rain making way for the sun as the game went in to the final quarter, Myler landed his second penalty to out the Saints back in to the lead.
With five minutes left on the clock the Saints won a strike against the head at a 22-metre scrum, allowing wing Diggin to go over in the corner with Myler adding the extras to deny the Chiefs a losing bonus point.
But Exeter, knowing that they needed to score twice, kept battling away and were rewarded with their second try as Shoemark burst clear and Baxter picked up at the back of a scrum to score his second try of the season.