Tries from prop Henry Thomas and centre Sam Tuitupou, along with two conversions and three penalties from Nick Macleod, gave Sale a commanding lead at the break in Friday night's Aviva Premiership clash.
It was just reward for their bold attacking approach, yet Exeter hit back with a 22nd-minute from Richard Baxter and were a side transformed after the restart and kept Sale scoreless during the second period.
After Thomas was sinbinned for Sale in the 55th minute, Exeter went on to claim a penalty try, which Argentine Ignacio Mieres converted to cut Sale's lead to three points.
The replacement fly-half then kicked a penalty to level the scores in the 72nd minute and scampered clear in the last minute after intercepting an errant pass from Tasesa Lavea to give the visitors victory.
Baxter, whose side had won at Edgeley Park last season, said: "The game looked to be getting away from us but credit to the guys.
"They got the try that stemmed the flow a bit in the first half, but really we were close to cracking.
"All we said at half-time was that we needed to do the things that re-established pressure.
"It was a massive challenge to our forwards but, to be fair, they cranked things up after half-time and forced some really good pressure at scrum time.
"We carried a lot better and got on the move a lot better and more accurately.
"We got the fundamentals right in the second half and that allowed us to play some pressure rugby."
In stark contrast to Baxter, Sale counterpart Steve Diamond could not hide his anger after his side frittered away their hopes of victory and were left with nothing more than a losing bonus point.
The executive director of sport, who also felt his side has been harshly treated by the officials during the second period, said: "We just didn't perform in the second half but full credit to Exeter.
"There was a lack of decisions going our way in the second half but you tend to get that if you're not being dominant.
"We seemed to get the rub of the green in the first half but in the second half we didn't get any decisions going our way, which is difficult to stomach.
"We talked at half-time about going out with the right mentality in the second half because it happened against Leicester and it happened again here.
"There are a lot of players who need to look at themselves because we've been beaten by a team on desire and decision-making."
The points were tied at 23-23 until Lavea's mistake allowed Mieres to race clear for the match-winning try in the last minute.
Diamond added: "He's an international player and you run the game out and get two points in this league.
"We go away with nothing - I'm embarrassed."
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