Solskjaer was subjected to calls for his sacking on social media after United needed a 5-3 penalty shoot-out win to edge past Rochdale after a limp 1-1 draw at Old Trafford.
Following Sunday’s chastening Premier League loss at West Ham, United were in trouble again when Rochdale’s 16-year-old Luke Matheson cancelled out Mason Greenwood’s opener.
United keeper Sergio Romero saved the day by denying Jimmy Keohane in the shoot-out, earning a last 16 trip to Chelsea.
It was a let-off for Solskjaer’s men and a further reminder of the need to make their chances count, having converted just one of their 31 shots.
“We go 1-0 up in many games. Today it’s again gone 1-0 up and you think ‘go on then, get the second’ which is the key now and that is learning curve for these boys as well,” said Solskjaer, whose side are a lowly eighth in the Premier League.
“We need to go for a second. At this club you don’t just sit back and hope that’s enough good. If you settle for good enough, that’s not what we want.”
Greenwood, 17, was one of the few positives after wriggling free to score in front of the Stretford End – just as he did in the Europa League against Astana last week.
“He has two great feet and took the penalty with his right, so he must be a nightmare for defenders having him one on one in the box,” Solskjaer said.
“He can go both ways, which is for me fantastic and you can just see he has got quality and he will get his share of games.”
Under-fire Solskjaer could have done with an easier draw than a clash with Chelsea, but United did beat them on the first day of the season and he said: “If you want to win it, you have to beat the best.
“We play them first game of the season this year. We played them in the FA Cup last year and it slots into a couple of weeks with loads of travelling.”
United will have four successive away games, with trips to Norwich, Partizan Belgrade, Chelsea and Bournemouth as October turns to November.
“That’s how it is. And we have got to make the best out of it – and I am sure we are because that’s the type of group this is, that’s the type of club we are,” Solskjaer said.
“We are not going to say ‘argh’, we’re going to relish it and look forward to it.”
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