Tevez put a difficult week off the field behind him as he set the tone for a comfortable afternoon with two first-half goals, either side of setting up Aleksandar Kolarov.
The Argentine completed his treble early in the second half and David Silva also got in on the act as City cruised through to the last four to keep alive their hopes of reclaiming the trophy they won two years ago.
Tevez could face up to six months in prison after he was arrested on Thursday night on suspicion of driving whilst disqualified, but he showed no sign of anxiety at the Etihad Stadium as he opened the scoring on 11 minutes.
Yaya Toure made it with a quite brilliant flick over the Barnsley defence. Silva was onto it in a flash, prodding the ball goalwards with the outside of his left foot. Luke Steele managed to push the effort onto a post but the rebound fell kindly for Tevez, who had the easiest of tasks to finish.
It was the signal for City to take complete control, outmanoeuvring Barnsley at every opportunity and, just before the half-hour, they struck again.
Once more it was a goal of classical efficiency. After a midfield skirmish, Edin Dzeko threaded a superb pass through for Tevez, whose cutback rolled to the back of the area, from where Kolarov drove it home.
City made it three in the 31st minute after another flowing move. Despite being surrounded by defenders not that far from the byline, Silva still managed to squeeze a pass to Tevez, who turned quickly before firing home from close-range.
City emerged for the second half with the same potency. Samir Nasri burst past former Blues trainee Kelvin Etuhu before finding Tevez, who delivered another expert finish.
Steele needed to make an acrobatic save to deny Dzeko, who was not far off with an audacious chip.
It was Silva who got City's fifth, though, finishing at the second attempt after Steele had initially denied him.
Barnsley's on-loan Manchester United midfielder Ryan Tunnicliffe tried to make an impression with a driving run through the City defence but he was denied by smart save from Costel Pantilimon.
James Milner smashed a shot against the woodwork late on as City had to settle for five, but on this form they will be difficult to stop and they will head into the semi-finals full of confidence.