Jones scrambled the ball over the line in first-half stoppage time after Motherwell had survived a decent spell of pressure.
The visitors were comfortable after the interval and Sutton removed any doubts over the outcome when he netted from close range.
Motherwell failed to hit the heights of their recent displays against Rangers and Kilmarnock but they lost both those encounters and manager Stuart McCall would have welcomed the scrappy win.
The victory came at a cost, though, with Keith Lasley and Steven Saunders both suffering injuries and Steve Jennings picking up a suspension.
McCall, who is set to complete the signing of Francis Jeffers despite playing down the chances of a deal for the former Arsenal striker on Friday, again freshened up his team.
Tom Hateley and Sutton were recalled and on-loan Blackburn defender Gavin Gunning was handed a first start in place of the injured Stevie Hammell.
David Mitchell comfortably held two Hateley free-kicks as Motherwell sought an early breakthrough and soon saved a Jones effort after good link-up play between the forward and partner Sutton.
Stranraer came into the game midway through the half and Steven Noble shot over from 20 yards after good skill, before Danny Mitchell blazed over after being afforded space on the edge of the box.
Gunning was booked for dissent while his fellow defender Shaun Hutchinson was cautioned for a crunching challenge and Stranraer forced another chance which Craig Malcolm volleyed just wide.
Jamie Murphy came on in the 31st minute for Lasley, who failed to shake off an early rib knock, and immediately injected some much-needed ingenuity into Motherwell’s play, setting up Saunders to shoot just over from long range.
Motherwell maintained the pressure but Sutton and Murphy could not make the most of Hateley set-pieces and Chris Humphrey could not quite turn a Jones cross into a team-mate’s path. However, Jones stabbed the ball just over the line after Saunders’ cross fell for him at the back post.
Stranraer right-back Grant Gallagher cleared and referee Charlie Richmond initially played on before assistant Tommy Johnston signalled for a goal.
Stranraer pushed forward at the start of the second half but Stephen Craigan blocked well from Kevin Nicoll and Motherwell proceeded to keep their Irn-Bru Third Division opponents at arm’s length without looking entirely convincing.
Jones ended a series of impressive runs with poor finishes and Murphy had a half chance when he ran on to Saunders’ clearance but Mitchell blocked the striker’s shot from a tight angle.
Jennings received a second booking of the competition for a mistimed tackle, which will see him suspended for the quarter-final.
Hateley came reasonably close with two free-kicks and his interception and through-ball set up Sutton, but the striker took too long to shoot.
Saunders pulled up injured before being replaced by Jonathan Page, who soon had a hand in Motherwell’s second in the 83rd minute.
Page fed Humphrey, whose low cross was forced home at the near post by Sutton, and the striker and Gunning both missed late chances to give the scoreline a flattering sheen.