Tevez was arrested for allegedly driving while disqualified on Thursday, an offence that carries a maximum six-month jail term, but he scored three times and made two goals at the Etihad Stadium as City joined Wigan in the last four.
Barnsley, who last reached the semi-finals in 2009, came to Manchester intending to defend en masse but their resistance was broken inside 11 minutes.
Yaya Toure found David Silva with a delightful flicked pass over the top and when Barnsley goalkeeper Luke Steele pushed the Spaniard's volley onto the post, Tevez was on hand to tuck the ball home.
The Argentine created his side's second goal in the 27th minute, tearing into the Barnsley box and rolling a deliciously weighted pass square for Aleksandar Kolarov, who found the bottom-right corner.
Tevez made it 3-0 four minutes later, gathering Silva's cut-back and spinning before beating Steele with a low shot from close range.
The hat-trick goal arrived five minutes into the second half and saw Tevez guide a low shot into the bottom-right corner after a gambolling run and cross from Samir Nasri.
Silva hit the hosts' fifth in the 65th minute, following in to score after Steele had saved his initial effort, with Tevez again supplying the assist.