With survival at stake, a solid first-half performance from Sale took them into the break 9-3 ahead through three Nick Macleod penalties.
Bath, who still harbour top-four ambitions, responded before the interval via Tom Heathcote's three-pointer.
Horacio Agulla's try then took the west country outfit into the lead, but Cueto's excellent finish sealed the win and took them six points clear of London Welsh, who play Gloucester on Saturday.
In snow-laden conditions it was the Sharks who had the first foray into the opposition 22 as Macleod's well-judged kick was taken into touch by Nick Abendanon.
Dave Attwood did break through the hosts' rearguard with a powerful run but, after that move broke down, the match remained attritional in the early exchanges.
That was until Abendanon's poorly directed clearance fell in the grateful arms of James Gaskell. Good hands then set up an excellent position from which the visitors were caught offside and Sale's stand-off put them 3-0 ahead after 21 minutes.
Steve Diamond's men were starting to get on top with their impressive driving maul to the fore, but it was Bath who had a chance to level matters when the home side were penalised for not rolling away.
Heathcote missed from the edge of the 22 but made up for that error soon after when he equalised from further out.
However, it was not long before the Sharks were back in the lead as Bath infringed straight from the restart.
Macleod converted the resulting opportunity and then added a further three-pointer just before the interval as they established a deserved 9-3 half-time advantage.
But Gary Gold's team had the wind at their backs in the second period and they immediately put pressure on the Sharks' defence.
It eventually paid off when Matt Banahan's surge and off-load found Agulla and the winger scampered over.
Heathcote converted to move the visitors into the lead for the first time and it looked ominous for the side second-bottom in the Premiership.
Nevertheless, Sale continued to show plenty of determination with their forwards particularly carrying well.
And they found a decisive score when a well-worked move saw Cueto collect the ball out wide and he showed express pace to break the gainline and touch down.
Although Heathcote reduced the arrears late on to one point, the Sharks held on.