They did concede two tries during that time - one of which was a penalty try - but they still stood tall and struck when the chances came their way.
Saints were strong in the scrum, sturdy in defence and spiky on the counter-attack.
So good were the black, green and gold that Exeter, the champions of England, looked to have run out of ideas.
Saints were 30-17 up heading into the final 10 minutes of the match, having scored three sublime tries.
Each effort, from Rob Horne, Ben Foden and Harry Mallinder, encompassed some eye-catching team play.
It was the kind of slick attacking showing that Saints produced during their four-match Aviva Premiership winning streak back in September.
And up front and in defence, they were much improved from the dark days of late 2017.
But sadly for Saints, the game just went on for too long for their liking and Exeter, who had made a series of handling errors, finally found their form.
Full credit to the Chiefs, who could have buckled at 13 points down, staring a fourth successive league loss full in the face.
They kept fighting and Saints were eventually undone, with the coaches ruing their side's indiscipline and some of the referee Christophe Ridley's calls.
Late on, a lack of astute game management - including a misjudged box kick - proved costly and there was also the odd handling error that didn't help the away side's cause.
They were so, so close to their first league win on the road since September.
But in the end it was agony rather than ecstasy.
Had Saints managed to stretch their lead by just one more try, they would have walked away with five points that would have kept them in the hunt for the top six.
But as it is, the aim of Champions Cup rugby coming to Franklin's Gardens next season is fading fast.
Saints can take plenty of positives from their performance at Sandy Park though.
And going forward, they should be all the better for it.
They are laying some foundations for next season and they can use the final six league fixtures to continue their rebuild during a tough campaign.
They would have loved to given their fans a long-awaited happy homecoming on Saturday evening.
It wasn't to be.
But any Saints fan in attendance at Sandy Park will certainly have been proud of so much of the performance, which hasn't often been the case away from home this season.
It is just a shame they couldn't celebrate the win they desperately wanted and, for so long, definitely deserved.
How they rated...
Returned from England duty and bagged himself a try, showing good awareness and desire to get up and support in a slick attack... 7
Will rue a late drop that gave Exeter the chance to launch a crucial attack, but the Samoan did little else wrong... 6
ROB HORNE - CHRON STAR MAN
A huge performance from the Australian centre, who scored one, set two up, launched numerous attacks and put his body on the line in defence... 8
A really impressive performance that included a brilliant break that eventually led to a try for Ben Foden... 8
Showed good speed to finish off the move for Saints' second try and he was unfortunate to be forced off early in the second half... 7
Produced a piece of brilliance in the build-up to Mallinder's score and this was easily his best performance since joining Saints... 8
Set the tone early on, racing around the pitch and keeping the tempo high for his team as they took Exeter on... 7
Was sin-binned during the first half after Exeter scored a penalty try, but the prop displayed his physicality again and did well in the scrum... 7
Another energetic showing from the hooker, who covered a huge amount of ground and was taken off just before Exeter scored their winning try... 8
Another decent day for the prop, who has been much improved in recent weeks and is looking solid at the moment... 7
Came into the side at late notice and certainly added plenty of hard work, defending superbly at times... 7
Captained Saints once again and he put everything he had into the game with Saints short of experienced second row replacements... 7
Another big display from the back row forward, who not only showed his determination in contact but also displayed his awareness with an assist for Horne's try... 8
Made a couple of crucial turnovers for Saints and is clearly now getting up to speed, showing why he was brought to the club... 7
Plenty of fight from the No.8, who scrapped for every yard once again and was unfortunate to be the man to pay the price with a sin-binning just before Exeter scored the winner... 7
Replacements (who played more than 20 minutes)
NAFI TUITAVAKE (for Foden 52)
Came into the game at a time when Saints were really giving Exeter a fright, but the Chiefs eventually recovered and he was left helpless as Don Armand sidestepped him for a try... 6