Exeter's future in the LV= Cup rests on the outcome of Sunday's match at the Brewery Field, where Harlequins need just a point against Ospreys.
The Chiefs' 28-19 victory, their first win after five defeats and a draw, denied Northampton a place in the last four with Bath topping Pool Four.
And Ricky Pellow, the Exeter skills coach, hopes Ospreys can cause an upset.
He said: "If they can do us a favour and beat Quins, and they don't get a bonus point, then we will go through [to the semi-finals] on try difference.
"It will be a nervous time watching the game tomorrow. We would like to have been an extra point up with a five-point win but any win in this competition, especially against Northampton, is a massive step forward for us as a club.
"We talk about moving forward each week and putting in the performances and this win shows we have moved on from last week at Bath. This week we created chances and took them.
"We talked all week about the intensity that we needed to bring to the game in our carries and intensity in our line speed. We didn't give them any chance to breathe and it showed when we created chances and took them.
"We had identified some areas where we could get some softer and weaker tackles to work an offload. We worked on it in the week and it paid dividends today.
"They came down with a strong squad and the momentum swung a little bit [when they used their bench] but we ripped it back off them and at the end we were pushing for the bonus point.
"Their driving lineout and their scrum were their strengths and I think we matched them at the scrum and the driving lineouts are hard to stop.
"We did it at times and they did gain a bit of momentum there and we fought really hard."
Exeter Chiefs went into the break 22-5 in front after tries from openside flanker Ben White, full-back Jack Nowell and wing Watisoni Votu, with fly-half Ignacio Mieres adding two conversions and a penalty.
Saints, who needed maximum points if they were to progress past the pool stage, squandered a couple of first half opportunities with their only reply coming five minutes before the break with a try from loosehead prop Alex Waller.
The visitors had the better of the second half with replacement scrum-half Lee Dickson crossing twice, both converted by replacement fly-half Stephen Myler, with Mieres replying with two penalties.
Northampton Saints director of rugby Jim Mallinder, whose side will now have to rely on a top four finish in the Premiership for a place in next season's Heineken Cup, said: "It's disappointing.
"Of course we knew that we need a bonus point and what you have to do when you come to a place like this is match teams,
particularly on the physicality side. They were more up for it as we slipped off tackles.
"I'm pleased when we came back in the second half when we were a lot more intent, particularly in our defence, we were driving them back and looking sharper in attack. But it was a little bit too late."
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