The Old Trafford club have had a few near-misses over the past couple of seasons, reaching multiple semi-finals under Ole Gunnar Solskjaer but failing to actually win any of the cup competitions.
They similarly came second in the Premier League this term, but were never truly in a title race which their rivals City had effectively wrapped up weeks ago.
Now, though, United face Villarreal in what looks an open affair in the Europa League final, with Greenwood saying United must seek to earn a trophy as the great teams of the past constantly did.
“To bring home a trophy is something the club needs, to bring home some silverware,” Greenwood said.
“We’ve not had any for a few years. To be a part of the team that actually does win a piece of silverware will be a good feeling.
“When you come and you’re in the academy, there’s pictures and photos on the walls of them back in the day winning the Champions League, the Premier Leagues, the FA Cups and that’s what it’s all about, really, for a football club. If you want to be the best football club in the world or in the country you’ve got to bring home the silverware and that’s something United have done over many years.
“But we’ve maybe dropped off a little bit throughout the recent years but hopefully we can after this one, if we win against Villarreal, that can be like a little stepping stone to get us started on where we want to be.”
On a personal level, Greenwood says that despite not hitting as many goals as might have been expected, he has made big strides this year.
That improvement has come in a tactical and a teamwork sense, where his fellow United players might appreciate his off-the-ball work more now as much as the fans appreciate his on-the-ball output.
“I think I’ve done better than last season,” he continued. “The performances when I [wasn’t] scoring, I still think I was playing well. I don’t think I was playing poorly.
“The final end product might not have been there and obviously towards the end of the season it’s kicked in, but I still feel like I’ve done a lot better than I had last year.
“My goals and assists might not seem like it but in my opinion — and maybe the players around me — I feel like I’m more of a player for the team now as well.”