Nathaniel Mendez-Laing scored in both halves - the first from the penalty spot - as the Bluebirds snatched a surprise 2-0 victory at Old Trafford, extending United's winless run to six matches in all competitions.
They end the season in sixth, a whopping 32 points behind Premier League champions Manchester City, and Solskjaer is aware that bridging that gap will prove a sizeable hurdle.
"We are five or six points behind third, fourth and fifth, that's realistically who we need to challenge for next year," he said.
"The players have been going through a tough period, good players who towards the end of the season are fatigued mentally and physically."
Solskjaer, whose side could start their 2019-20 campaign in the Europa League towards the end of July depending on the outcome of the FA Cup final, has already set his sights on a top-four spot and a domestic trophy next term.
He added: "We're used to challenging for the Premier League and that's not going to be sorted quickly. Realistically you're not going to catch however many points - we're 32 points behind the top teams.
"We've got to realistically challenge for the top four and another trophy.
"Last time we were in the Europa League with Jose (Mourinho) and loads of these players, we won it. The FA Cup and the Carabao Cup, they're trophies we've got to start the season aiming to win."