They have strengthened a squad that finished fourth in Serie A last season by signing veteran Uruguayan worldwide defender Diego Godin and promising midfielder Nicolo Barella.
Lukaku's departure leaves United with just three established forwards - Rashford, Anthony Martial and Alexis Sanchez - in addition to youngsters Angel Gomes, Mason Greenwood and Tahith Chong.
With United unable to sign a replacement, the pressure will be on 21-year-old England forward Marcus Rashford to deliver the goals in a more central role than he has previously operated in. Signed from Swansea City in the summer, the Red Devils acquired a winger who recorded the third-highest fast breaks per 90 minutes (0.25) in the Championship of any player in 2018/19.
Lukaku joined the Serie A club on Thursday, the Premier League's transfer deadline day, but there was no new arrival to take his place at Old Trafford before the window shut.
There was still time for substitute Daniel James to complete a dream debut with United's fourth in the 81st. "He was always my number one target", Solskjaer said.
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"When you haven't got the correct answers from the players you wanted to have it's like: 'hmmm there is doubt.' Most of the time there's no doubt so then you'd rather trust the ones we have".
It's believed United were targeting Juventus forward Paulo Dybala, but that deal looks unlikely now Lukaku has moved to Milan, rather than Turin in a swap deal. I think Inter are heading in the right direction and I needed a new challenge. He is either a player who starts and plays all the games, and he will score goals, but he is not a player to come in and out.
"Mason Greenwood's pathway would have been a lot more different if we had another forward there".
"I do expect him to stay, there's a striker there we think is going to be able to score some goals".
Arsenal will allow more players to leave the club before the transfer window shuts for the rest of Europe's major leagues on September 2, manager Unai Emery said on Friday.