Cueto was making his first appearance of the season for Sale after World Cup duty for England and was instrumental in a decisive second-half try for Johnny Leota.
Leota's 56th-minute score followed first-half tries from Will Addison and Alasdair Dickinson as Sale came from 14-0 down to lead 15-14 at the break.
Yet Rory Clegg kicked two penalties to put the visitors back in front and Stegmann, 22, completed his treble late on to hand Conor O'Shea's men a second straight win in the competition and Sale a second successive defeat.
Quins led in the seventh minute when Richard Bolt found Clegg and his neat offload found Stegmann advancing forward at pace.
The wing showed impressive pace to burst past a clutch of Sale defenders inside the home 22-metre line before crossing for a fine individual effort.
Clegg added the extras before Cueto showed intelligence to take a quick penalty tap and embark on a marauding run down the left flank.
It came to nothing but set the tone for period of sustained pressure on the visitors' line which saw Sale kept at bay by some desperate defending.
However, the Sharks should have touched down midway through the opening period after some fine handling from Nick MacLeod and Luther Burrell created a clear opening for Cueto.
The England wing burst onto Burrell's short offload at pace but spilled the ball as he attempted to ground it and the opportunity was spurned.
Harlequins then lost centre Tom Casson to injury in the 25th minute after he took a heavy knock before being carried off on a stretcher and replaced by Matt Hopper.
But two minutes later, Quins struck again as Tom Williams picked up a wayward pass from a Sale defender and scampered clear unopposed from halfway and Clegg added the extras.
Yet Sale refused to lie down and MacLeod got them off the mark with a penalty before a fine break inside the right channel culminated in Addison outpacing the Quins defence to cross the line.
MacLeod added the extras and Sale struck again on the stroke of half-time when a pass was fired out from the base of a scrum and Dickinson took it to squeeze over in the left corner.
MacLeod failed to convert and four minutes after the restart Stegmann plunged over in the right corner for his second try and Clegg applied his third conversion.
MacLeod then responded with his second penalty and Clegg landed a penalty before MacLeod found Cueto in the 56th minute and his neat pass sent Leota clear.
MacLeod added the extras to put Sale 25-24 ahead but he then sent a penalty wide and in the 68th minute Clegg booted a penalty from halfway to edge Quins 27-25 in front.
Clegg then added another penalty before Stegmann burst clear down the right flank for his hat-trick and Clegg added the extras.
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