Cleverley - who had missed a great chance in the first half - slotted home a cool finish to double United's lead, which came from Anderson's opener late in the first half.
Papiss Cisse pulled one back for the visitors only seconds after coming off the bench, making for a tense finish that saw the Senegal striker hit the bar with a spectacular volley, but a youthful United - who made 11 changes from the weekend win over Liverpool - held on for victory and a place in the last 16.
More good news for Sir Alex Ferguson were the returns of Wayne Rooney, who had missed three weeks with a nasty leg injury, and Darren Fletcher, who made his first start this year due to illness.
Holders Liverpool beat West Brom 2-1 in the late kick-off, with Nuri Sahin getting both their goals, his first for the club. Gabriel Tamas had opened the scoring for the Baggies after Reds keeper Brad Jones failed to hold on to his effort, while Baggies keeper Ben Foster was also at fault for Sahin's equaliser.
The winner came late on as the Turkey international, on loan from Real Madrid, converted a fine cross from fellow summer signing Oussama Assaidi.
Arsenal beat League One Coventry City 6-1 as Olivier Giroud scored his first goal for the club to put his side ahead. Alex Oxlade-Chamberlain, Andriy Arshavin, Theo Walcott (2) and Ignasi Miquel completed the rout.
Tottenham won 3-0 at Carlisle through goals by defender Jan Vertonghen, Andros Townsend and Gylfi Sigurdsson.