Sale are scrapping for their Premiership survival and were outplayed in the opening period as Exeter dominated.
A try from Watisoni Votu and eight points from the boot of Gareth Steenson eased them into a 13-5 half-time lead.
Dan Braid gave the Sharks hope when he went over, but Steenson extended the advantage early in the second half.
However, the home team fought back after that and Cueto touched down for his 76th domestic score before Rob Miller kicked a penalty.
And the Manchester outfit gained a vital victory through Danny Cipriani's three-pointer and Nick Macleod's drop-goal.
Despite enduring a disastrous season, the hosts came into the game following back-to-back wins in the LV= Cup to give them a home semi-final against Saracens.
It was a much-needed confidence boost for the club, but it was the visitors who started the brighter.
Rob Baxter's men have themselves found it difficult in recent weeks having failed to win in their previous three Premiership games.
However, they were far more inventive and coherent early on. And the Chiefs should have opened the scoring when the ball was shifted wide to Jack Nowell but, with the line at his mercy, the winger knocked on.
Exeter did eventually take the lead when an infringement led to Steenson kicking them ahead.
The Sharks were the next to threaten after being awarded a penalty from the restart. Instead of going for the posts, they went for the corner which proved to be the correct decision when Braid touched down from a driving maul.
Despite that setback, the away side continued to control proceedings with Steenson and Will Chudley dictating operations from half-back superbly.
And their fly-half deservedly put the Chiefs back in front with a second three-pointer.
Sale's director of rugby Steve Diamond targeted this encounter as a must-win match, but his charges were slack in comparison to Exeter's typically energetic display.
It was, therefore, no surprise to see the visitors extend the lead when a period of pressure ended in a try.
Sireli Naqelevuki made a rumbustious charge through the middle before the ball was spread wide to Jason Shoemark. The Kiwi made further inroads and the gargantuan Votu capped off a fine first half performance by touching down.
Steenson added the extras to extend the visitors advantage to 13-5 at the break.
The second period initially continued in the same vein with Exeter looking the livelier and their fly-half duly kicked them further into the lead.
But, to their credit, Sale stormed back. An excellent move which involved forwards and backs alike ended in Cueto going over for his record-breaking score, Rob Miller converting.
Miller then reduced the arrears to one point with 20 minutes remaining.
And when Kai Horstmann was sin-binned for persistent team infringements, Cipriani took the Sharks into the lead.
Macleod then added a drop goal and they held on despite Nowell having a try ruled out late on.