Sanchez followed Lukaku to Inter Milan, securing a season-long loan move to the Serie A side – managed by former Chelsea manager Antonio Conte – this week.
Belgian frontman Lukaku joined the same club earlier in August in a £73million deal.
Those departures leave United with just Marcus Rashford, Anthony Martial and 17-year-old Mason Greenwood – with just one Premier League start to his name – as the club’s recognised centre forwards.
United were linked with a handful of forwards before the transfer window for English clubs closed on August 8, including Juventus duo Paulo Dybala and Mario Mandzukic.
Dybala was tipped to join the Red Devils in a swap deal that would see Lukaku go the other way, while Mandzukic was mentioned as a separate target at around £13m.
But neither deal transpired with Solskjaer instead settling on just three summer signings in Daniel James, Aaron Wan-Bissaka and Harry Maguire at a combined cost of near £150m.
And the Norwegian coach has admitted United are lacking in numbers at the top end of the pitch but says he did not want to panic because he could not find the ideal player to give him an extra option.
“As I said on the way back from tour, we were still looking for one replacement for Romelu if he went,” said Solskjaer.
“We are one down to be fair, but we didn’t just find the right one. We didn’t find the answer that we wanted.
“I’m still very happy. We might be one down in the attacking line-up, but we just didn’t get the right answers we wanted.”
United have started all of their league three matches so far this season with both Martial and Rashford starting.
And against Wolves and Crystal Palace that has left only Greenwood as an option to lead the line from the bench.
But Solskjaer felt that he simply could not guarantee Sanchez – who returned just five goals in 45 games for United – consistent game-time and so that is why he sanctioned his exit.
“I had a chat with him and for me if I can’t guarantee regular football for Alexis Sanchez, it’s better for the two of us that he goes and plays in Italy and shows what we know he’s capable of,” he added.
The United boss is also full of confidence that Martial and Rashford can carry the load, despite the question marks around the pair, as they seek the goals to fire them back into the top four and into next year’s Champions League.
Martial’s best ever Premier League tally stands at 11 while Rashford has never bettered 10, although the pair both have two apiece already after three games this term.
Solskjaer said: “Yeah, if I did not think they had enough I would not just have them. We would have signed someone else or kept someone else. We believe in them definitely.
“I think both of them enjoy responsibility and enjoy the trust, so it’s give and take here. We trust you, but you’ve got to give us back.
“They have shown me they are ready for that.”
United take on Southampton in the weekend’s early kick-off on Saturday (12.30pm) but will be without Martial at St Mary’s due to injury, with left-back Luke Shaw also absent against his former club.
Solskjaer’s side are without a win since thrashing Chelsea 4-0 on the opening weekend and are seeking to end a five-match winless streak on the road.