The victory moved Chelsea into fourth place and means a win at Nottingham Forest on the final day of the season will guarantee at least a top-five finishâenough to secure their return to Europe's top competition after a two-year absence.
The Blues could also end the campaign with silverware if they beat Real Betis in the UEFA Conference League final.
Despite the result, Maresca urged for patience with his young squad, highlighting the contrast between the current Chelsea project and previous title-winning teams.
"For me, the problem is that people, especially the media, still think Chelsea are the same Chelsea as years ago," he said. "We had six academy players in the squad tonight. Years ago, if Drogba was missing, it was Costa or Torres. This is a different Chelsea."
He added: "We are moving in the right direction and hopefully we can bring the club back where it deserves to be."
Chelsea, who have spent over one billion pounds on new players under their American ownership group, dominated much of the contest but struggled to convert their chances.
Noni Madueke wasted an early opportunity, and Reece James struck the post before the breakthrough came with 20 minutes remaining. James turned sharply on the right and delivered a superb cross that Cucurella met with a powerful header beyond Andre Onana.
Manchester United had threatened earlier, with Harry Maguire's finish ruled out for offside after a tight VAR check, and Mason Mount and Bruno Fernandes both missing chances. United pushed for a late equaliser, but Chelsea goalkeeper Robert Sánchez denied Amad Diallo at the near post.
A penalty awarded to Chelsea for a challenge on Tyrique George was overturned by VAR, but the Blues held on. The defeat left United winless in their last eight league games, their worst Premier League run in history.
United manager Ruben Amorim, who fielded a strong team just days before their Europa League final in Bilbao, now faces growing pressure. Without success on Wednesday, United's season could be remembered as a major failure.