United lost 2-1 on the night at Hull City but edged through to a Wembley place on aggregate in midweek.
It gives Jose Mourinho the chance to build on Louis van Gaal's FA Cup success last term and claim back-to-back showpiece final triumphs when his side take on Southampton next month.
But with a seemingly favourable path to the last-16 against the Latics next on the agenda, Jones knows United must keep pushing for success with their prospects of a top-four finish in the Premier League once again in the balance.
"It's not enough just to win a cup. We set our standards high and this club has a history of winning things," Jones told reporters, before recalling how last term was broadly considered a failure for United despite them managing to pick up a piece of silverware.
"We are under no illusions that we have no given right to win things, but winning the FA Cup is not enough.
"That result in midweek was not what we wanted. We are delighted to be through to the final and to go to Wembley, but we weren’t happy with ourselves.
"That took the gloss off it a bit and it was a disappointed dressing room - but maybe when we look forward to the final it will be better.
"We know it wasn’t good enough, especially in the first half. We had to dig deep, but we conceded another sloppy goal.
"It's a mark of how well we have done that we are so disappointed with that result.
"We set our standards high and we were proud of the unbeaten record we’d been on.
"It takes some character from players to go on an unbeaten run as long as that and to throw it away was disappointing. Personally, I was devastated by it."
Aside from competing for honours domestically, Jones hopes United can establish themselves as this season's cup specialists by having a tilt at the Europa League.
"We’re in every cup competition and there is no reason why we can’t reach the Europa League final, too," he added, with Mourinho's men hosting Saint-Etienne next month.
"But we need good performances and everyone in the squad pulling in the right direction and battling for each other.
"The Europa League is a genuine target for us. We want to be in the Champions League, but if it’s not to be we are still in Europe now and there are some good teams in the competition."