Goals by David Silva and Sergio Aguero helped City brush aside their opponents challenge.
City opened the scoring as early as the third minute after Yaya Toure’s pass was headed back across goal by Pablo Zabaleta and with Silva applying the finishing touch.
City had many chances to go ahead with Silva failing to convert a similar goal opportunity in the second half.
Middlesbrough’s Brad Guzan made four important saves in the space of 10 minutes in the second half, twice from Leroy Sane and also from Silva and Aguero.
Aguero finally beat Guzan in the 67th minute when he turned in Sane’s excellent cross for the second goal.
In other quarter final clashes, Arsenal take on minnows Lincoln in what is seen as a tricky encounter for the former champions.
The match follows a miserable week for Arsenal, which lost 3-1 to Liverpool in the Premier League and 5-1 to Bayern Munich in the Champions League, which was the club’s heaviest defeat in 1998.
Arsenal manager Arsene Wenger has had to face protests from Arsenal fans who are calling for his resignation.
“I have worked very hard for 20 years to make our fans happy and when you lose the games, I understand they are not happy,” Wenger said.
“I do not want to judge the protests, I am not able to. I don't work for my image, I work for this club with full commitment.”
On Monday, Manchester United take on Chelsea in a high-voltage clash.
United were thrashed 4-0 at Chelsea in October and United manager Mourinho was taunted by his old supporters on his first return since his sacking as Chelsea boss last season.
Asked if Mourinho's League Cup winners want revenge, United forward Marcus Rashford: “Yeah, definitely. They’ve got one over us. We're on a good run of form and so are they, so it'll be a good game.”