After 280 minutes without a goal, United finally found the net when Jones delivered a powerful finish to Robin van Persie's corner.
It was enough to see off a dangerous Shakhtar Donetsk outfit, who had more shots than their hosts, on and off target, but failed to condemn Moyes to a third straight home defeat.
A scrappy, stuttering and, at times, unconvincing performance will not be enough to silence United's critics - in his half-time analysis on ITV former skipper Roy Keane declared Ashley Young to be not good enough to play for his former club.
However, in what has become a miserable debut campaign on the domestic front, Moyes at least has a European tie to look forward to next spring, with many of the tournament heavyweights being avoided when the last 16 draw is made next Monday courtesy of them finishing top of Group A, unbeaten.
And a reminder that Jones' only other European goal came against Basle two years ago in a game that saw United exit the competition at the group stage should at least check those who have conveniently forgotten life under Sir Alex Ferguson was not always straightforward.
Whilst Moyes said in the build-up he took complete responsibility for United's downturn in fortunes, his players have been the ones out on the pitch.
As has often been said, they won the Premier League by 11 points last season, so their collective ability should be enough to shoulder some of the burden themselves.
When Jones surged onto Rafael's return pass and brought a close-range save out of Andriy Pyatov it did seem United were in the mood to at least maintain their impressive European form.
Pretty soon though, the anxiety that has enveloped them during those back-to-back home losses to Everton and Newcastle returned.
Taison's through-ball presented Alex Teixeira with a clear run at goal and his flick rolled narrowly wide of the far post.
Though Adnan Januzaj did bring a decent save out of Pyatov, it was the start of an extended period of Shakhtar pressure, in which United, clearly lacking in confidence, struggled to string more than a couple of passes together.
A lone cry of "get a midfield in there" could be heard about the general groans and not even Wayne Rooney curling a free-kick into the side-netting could counter the growing feeling United were in perilous danger of falling behind.
Fred should have turned that fear into reality after Teixeira had robbed Rio Ferdinand, who had a shocker, but despite the ball being rolled invitingly into his path, the shot lacked power and did not threaten the home goal.
A couple of good opportunities for Young came and went.
The winger lobbed Pyatov with the first but as the ball dropped narrowly wide, Young would have known if he had elected to square for Rooney, the United skipper would have had a tap-in.
Young also drove wide after collecting Jones' crossfield pass, prompting Keane's latest critical assessment of his former club during his half-time analysis.
Rafael blocked a Vyacheslav Shevchuk effort at the start of the second-half before the Shakhtar midfielder saw another attempt flick off the outside of a post after taking a slight deflection off Jones.
In response, Giggs meekly fired at goal following decent approach play by Young and Shinji Kagawa. Young's long-range curling effort to the far post dropped narrowly wide and Rooney's ambitious chip brushed the roof of the Shakhtar net.
Van Persie's introduction brought added potency to the home attack and the Dutchman was a key figure in their opener, albeit by taking the corner Jones lashed home.
The relief that rippled round the stadium as Jones' volley hit the net was palpable and whilst Shakhtar did try to respond, they were the ones who lacked conviction as Bayer Leverkusen's win away to Real Sociedad dumped them into the Europa League.