United crashed out of Europe's premier club competition on Wednesday night when they were beaten 3-1 by holders Bayern Munich, with the quarter-final tie lost 4-2 on aggregate.
As David Moyes' men sit seven points adrift of fourth place in the Barclays Premier League with five matches left, it would take a minor miracle for the team to qualify for the Champions League again next season.
That means United will miss out on playing in the tournament for the first time in 19 years.
And that does not impress Jones, who wants more nights like the dramatic one that unfolded at the Allianz Arena on Wednesday evening when United sniffed victory after going ahead in the second half.
"It's hugely important we get back in it as soon as possible," the United defender said.
"Manchester United need to be in the Champions League.
"Nights like these do not come around too often. They are the ones you look back on and think - 'what a time that was'.
"You want to play against the best and you want to test yourself against the best."
United have lost 10 league games already this season in what has been a woeful debut campaign for Moyes.
They will end the season with their worst points total in Premier League history, but midfielder Michael Carrick is confident that the squad are capable of finishing in the top four next term.
"Hopefully it will just be one year (out of the Champions League)," the England international said.
"We believe it will be. We are confident we be right up there challenging next season."
United were 11-1 outsiders to win on Wednesday night, but the travelling fans started to dream of another historic European night for their team when Patrice Evra beat Manuel Neuer with a fantastic 57th-minute strike.
But the goal only served to jolt Bayern into life. Just 22 seconds after the restart, Mario Mandzukic scored before Thomas Muller and Arjen Robben wrapped up the win for the reigning European champions.
Jones admitted the defeat was hard to take.
"I am gutted," said the centre-back, who had to deal with wave after wave of attack from Bayern over the two legs.
"This wasn't the way we wanted it to end. I thought we more than held our own against them tonight.
"We didn't disgrace ourselves against a really good team but we are still disappointed."