Cleverley's goal steered them towards a 2-1 Capital One Cup victory over Newcastle at Old Trafford.
But earlier, Cleverley had been sent through by Javier Hernandez when the contest was still goalless and amazingly, with only goalkeeper Rob Elliot to beat, Cleverley fired wide.
Although Anderson fired home an excellent 25-yard effort to put United ahead at the break, Rooney confirmed Ferguson was not happy.
"The manager wasn't too pleased with Tom because of that miss in the first half," said Rooney.
"But it was a nice goal by him and a great strike by Anderson.
"I am delighted for the two of them, especially Tom because it was his first one."
The prize for victory was a dubious one as it set up a fourth round encounter with Premier League pace-setters Chelsea at Stamford Bridge next month.
It represents a major test for United, not least because of their growing number of injured defenders, Nemanja Vidic and Phil Jones will both still be sidelined for certain.
The departure of Alex Buttner before the end of Wednesday night's contest was another worry for Ferguson.
However, there was consolation in the performances of youngsters Scott Wootton and Michael Keane, who until the final 20 minutes when Papiss Cisse scored a minute after his introduction as substitute, kept Newcastle at bay in impressive fashion.
"For two young centre-halves playing against top strikers they were fantastic," said Darren Fletcher, who was making his first start after 10 months out combating a chronic bowel condition.
"They were talking to me the whole game."
Keane went into the match with just one substitutes' appearance under his belt, whilst Wootton was making his debut.
Marnick Vermijl was also handed his debut at right-back, whilst Robbie Brady and Ryan Tunnicliffe - whose dad landed a hefty wager at 100-1 that his son would play a first-team game for United, struck when the midfielder was only nine - were also given their first experience of senior combat.
"Newcastle had a stronger team than us physically but we played some fantastic football and we deserved to win," said Ferguson.
"We want to progress as far as we can in this competition and will do our best to do that."
Newcastle boss Alan Pardew was disappointed with the outcome, believing his side had done enough to at least take the tie into extra-time.
However, their best chance disappeared when Cisse's overhead kick came crashing back off the crossbar near the end.
"There are a lot of positives to take out of tonight [Wednesday]," he said.
"The media can't hang that tag round Papiss' neck that he has not scored anymore.
"That goal means everything to him. There is nothing in his work-rate that has changed. Sometimes a goal can make a difference.
"You could see that little bit of exuberance return to his game with the overhead kick. I thought that was in."