The 71 year-old Scot was described by his assistant manager Mike Phelan as "too distraught" to take interviews after the defeat to Jose Mourinho's team, which saw United exit the competition 3-2 on aggregate.
However, with an FA Cup tie against old foe Rafa Benitez and Chelsea up next, Sir Alex says that they will recover well.
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"When you're at a club for a long time there are always damp moments and dark days. In general we recover very well and we'll do that again."
The FA Cup quarter-final will be the first time that United have faced the west London club since Benitez took over as interim manager.
Earlier this season, in the aftermath of Roberto Di Matteo's sacking, Sir Alex said that the former Liverpool boss was "very lucky" to manage another Champions League-winning team, after he took over at Inter in 2009.
Since then, however, the Spaniard has failed to win over a sceptical fan base, struggled to maintain the club's title ambitions, and criticised the Blues' heirarchy for naming him 'interim manager'.
But Sir Alex has insisted that he is not interested in the troubles of his old rival, and claims that he is instead remaining focused on his own affairs.
He added: "I'm not going to kick anyone when they're lying down. It's not my style.
"My concern is Manchester United. What happens at Chelsea doesn't really affect me. Chelsea have a terrific record in the FA Cup, probably the best of any team in the last decade.
"It doesn't matter when you play Chelsea or who's in charge, we'll have a hard game."