The Red Devils were well off the pace in the Premier League, finishing 24 points behind champions Chelsea, only to redeem themselves with victory over Ajax in the Europa League final earlier this week.
United therefore ended the season with two major honours, with that success in Stockholm adding to their EFL Cup joy, and Lingard insists that he and his teammates have proved that they are "capable of anything" in the years to come.
"I think finishing inside the top four and winning the league was our ambition at the start of the season," he told the club's official website. "We know we've got the players capable of winning the league but it didn't happen – we've had to play through the Europa League to get to the Champions League.
"Individually, we could have done better and as a team we could have done better, but football is all about learning and you're always going to learn with every game. We'll take this experience into next season. It shows we're capable of anything.
"There have been a lot of positives this season; the manager has got us well drilled and if we take that into next season with more goals and turning draws into wins, I don't see why we can't be challenging for the title.
"If we'd managed to convert those draws at home into wins we'd have been up there with Chelsea and Tottenham, so it's all there for us, but it's just about doing it on matchday."
United also won the Community Shield at the start of the season thanks to a 2-1 win over Leicester City.