Wells helped himself to a double either side of Jamie Paterson's fine finish as the visitors, under the guidance of caretaker manager Dean Holden, cruised to a 3-1 victory at the Riverside Stadium and secured back-to-back wins for the first time since February.
On an afternoon when the Teessiders paid tribute to former manager Jack Charlton before kick-off, they huffed and puffed and were unable to find the cutting edge they needed to repeat their midweek success at Millwall until Britt Assombalonga's late consolation strike, and they remain perilously close to the bottom three.
They were undone within six minutes of kick-off when Andreas Weimann pulled the ball back to Wells, whose instinctive curling left-foot effort beat keeper Dejan Stojanovic and went in off the post.
Things might have been even worse for the home side three minutes later had Stojanovic not got down at his near post to turn away Famara Diedhiou's skidding strike after he had cut inside from the left.
Saville was himself denied by the crossbar after keeper Daniel Bentley just managed to get his fingertips to the ball after the midfielder had met Ryan Shotton's long throw, although Stojanovic had to save from wing-back Jack Hunt at the other end after he had been played in by Paterson.
However, it was Paterson who doubled the visitors' advantage in emphatic fashion three minutes before the break, rounding off an impressive counter-attack by Hunt and Weimann after Bentley had plucked Paddy McNair's corner out of the air.
The pattern of the first half continued after the break with Boro pressing, but the Robins threatening and Tomas Kalas was left with his head in his hands after failing to make meaningful contact with Paterson's 55th-minute free-kick in front of goal.
City were home and dry with 11 minutes remaining when Wells stabbed past Stojanovic after Fry had failed to cut out Weimann's through-ball, and although Assombalonga reduced the deficit three minutes later, there was no way back.