The England striker limped off with a groin problem during the second half as United’s lack of options in attack were laid bare.
With Anthony Martial still injured and youngster Mason Greenwood ill, Jesse Lingard had to fill in up front against the Hammers.
The winger may also find himself doing so against Rochdale in the Carabao Cup on Wednesday and, more worryingly for Solskjaer, when Arsenal visit on Monday week.
There could also be repercussions for England, who face Euro 2020 qualifiers against the Czech Republic and Bulgaria next month.
Rashford will have a scan on Monday but Solskjaer admitted: “He’ll probably be out for a while.”
The 21-year-old’s absence will raise more questions about United’s summer transfer business, when they failed to bring in a replacement for Romelu Lukaku.
Of just as much concern for Solskjaer is their away form. United have not won on their travels in the Premier League since February – a run of seven matches equalling an unwanted club record.
The Norwegian’s predecessor Jose Mourinho stuck the boot in on Sky Sports, claiming he saw no improvements from last season.
“He’s got the right to his opinion,” said Solskjaer, before pointing out Mourinho’s United also lost at West Ham last year.
Andriy Yarmolenko broke the deadlock with a peach of a strike just before half-time, and Aaron Cresswell finished United off with a stunning late free-kick.
“It was a game that could have gone both ways,” Solskjaer said.
“The key moments, they grasped and we didn’t. I have to say two fantastic finishes from two good left feet. And we didn’t take our chances when we had them.
“It’s one of those moments. We’ve historically had loads of centre-backs struggling with injuries over the years.
“At the moment it’s the strikers. That’s just how it’s gone. Hopefully Mason and Anthony will be back for the Arsenal game, and at least one of them for Rochdale.”
Life looks far brighter for West Ham, whose captain Mark Noble this week claimed they could even break into the top four this season.
They are certainly heading in the right direction after a hard-fought third win in four matches, and a third consecutive clean sheet.
“I’m very happy about all of today,” manager Manuel Pellegrini said. “The way we played – I always talk with the players that we must have the mentality to play the same way we work during the week.
“It doesn’t matter if we play United or against every big team. We must trust that we are able to beat them.”
Provided by Press Association Sport
Pep Guardiola insists Manchester City’s continued pursuit of goals against Watford was a sign of respect rather than an act of humiliating their opponents.
City thumped Watford 8-0 on Saturday – their first five goals coming inside the first 18 minutes.
Asked about the issue of ‘humiliating’ the Hornets in the second half, Guardiola said: “The best way to humiliate the players, not respect the players, is second-half (do) stupid things and play for themselves and make some fashion movements.
“That is the worst thing to do that and that’s why it didn’t happen. So at the end, we were respecting incredible the opponents, that’s why we play in that way because we respect the opponents with or without the ball what we have done.
“And of course, we play for the spectators who come here to watch the game. So I don’t have any doubts about that. I would not liked to have watched or to see my team in the second half play just how ‘Watford they are and it isn’t necessary to run much because we are winning 5-0’.”
Nicolas Pepe is doing individual training work to help him adapt to life at Arsenal.
The Ivory Coast international joined from Lille over the summer having scored 22 goals in Ligue 1 last season.
Pepe has since started three of Arsenal’s six games at the start of the campaign and is yet to show why the Gunners paid a reported £72million to bring him to the Emirates Stadium.
Gunners head coach Unai Emery has used the 24-year-old as a second striker alongside Pierre-Emerick Aubameyang in two games, a 3-1 defeat at Liverpool and last week’s 2-2 draw away to Watford.
Now the Spaniard admits he and his coaching staff are working closely with Pepe to speed up his adjustment in the hope he can impress against Aston Villa on Sunday.
“He needs to adapt,” said Emery. “He is playing well but progressively we know there is going to be more from him to help us and I want him to take the last step quickly in adapting for us.
“We are working with him – after training we are watching individual videos with him.
“He is playing well but also we need in our organisation to be strong with every player to improve a lot.”
Emery and his players faced plenty of criticism after letting a two-goal half-time lead slip at Vicarage Road – where they were lucky to leave with a share of the spoils.
But, after making seven alterations, Arsenal impressed in a 3-0 Europa League victory over Eintracht Frankfurt on Thursday night.
Former Paris Saint-Germain boss Emery concedes he ultimately has to deliver for the club’s supporters but insists he is never too reactive to a result.
“When we are winning I stay calm, when we are losing I stay calm,” he added.
“I want to win more than lose and really I want to be consistent in our way and in our steps ahead. We know each match is very important because we have a lot of supporters in London, England and also worldwide.
“We know our responsibility and we know we want to give them very good moments, we want to give them special moments and feel connected with us because it is not only how we can win, it is also how we can play and how we can give them our energy, our capacity, our skills and enjoy each match with us.”
Pepe came off the bench in Frankfurt and is likely to be recalled to the starting line-up against Villa as Alexandre Lacazette remains sidelined with an ankle injury.
Fit-again Rob Holding will be pushing for his first appearance since December 2018 while full-backs Hector Bellerin and Kieran Tierney – signed from Celtic in the summer – both played a little over an hour in an under-23 fixture against Wolves on Friday night as they step up their respective comebacks.