The defender-cum-midfielder was United's match-winner in their final European game of the year, as they beat Shakhtar Donetsk 1-0 at Old Trafford on Tuesday.
United finished top of Group A, and will be aiming for a long run in the competition when it starts up again at the last-16 stage in February.
But their form week-in and week-out in the Premier League has been far short of expectation so far, with new manager David Moyes struggling to get the best out of the championship-winning team he inherited from Sir Alex Ferguson.
"We know we've had a disappointing couple of weeks but probably we can start going on a run now," Jones said on ITV1. "Hopefully we can kick on from tonight. I know we've said that in the past but it's really time we did start kicking on."
United started slowly and faced plenty of pressure in the first half, before improving after the interval.
"I think we had to show a bit more desire, a bit more belief in the performance, and I thought we did that eventually. We got to grips with it in the second half and created a lot more chances," Jones said.
A half-time prompting from Moyes amounted to a reminder of the basics, as Jones explained he told the players "just to pass the ball better, give each other nice passes and create chances".
"I felt we did that in the second half," Jones said.
Asked if it was a timely win for a team coming off two back-to-back home Premier League defeats, Jones said: "It was. We needed it."
The win came as a relief to Moyes who has not tried to downplay his new side's struggles.
"We got to the top of the group and that's what we wanted," he said.
"We wanted to win it and it helps us improve our form. We didn't play that well in the first half and they had two or three opportunities. We got better in the second half and were much better.
"I didn't think we were close enough to them (in the first half) and gave them too many opportunities. We were fortunate not to be a goal down."
Attention will now turn to the draw for the knockout rounds and Moyes thinks United should be aiming high.
"I think a club like Manchester United has to try and win the trophy," he said.
"That's the job and the teams in the cup are very good. We need to get better as it's a tough competition. Let's take it as it comes and see how we get on."