Tevez, held by English police on Thursday for allegedly driving while disqualified, took his goals tally for the season to 15 in all competitions with clinical finishing against the Championship club.
Wigan had earlier stunned Everton 3-0 at Goodison Park to reach the last four of the cup for the first time in their 81-year history.
Roberto Martinez's side, facing another Premier League relegation battle, were underdogs but effectively settled the contest with three goals in four amazing first-half minutes from Maynor Figueroa, Callum McManaman and Jordi Gomez.
The semi-finals will take place at Wembley on the weekend of April 13 and 14.
Tevez, 29, justified his manager's faith in him at the Etihad Stadium, scoring in the 11th, 31st and 50th minutes before being substituted 14 minutes from time.
"If it has this effect on him, I hope the police can stop him every day," Roberto Mancini was quoted as saying by the BBC.
Italian Mancini is aiming to guide City to their second FA Cup triumph in three seasons and their sixth in total.
Barnsley were outclassed throughout and never looked like even challenging their opponents once Tevez put the Premier League champions ahead with a tap-in after keeper Luke Steele had pushed David Silva's flick on to the post.
Tevez provided the second for Aleksandar Kolarov, who finished with an unerring, low shot across Steele, and Tevez made it 3-0 before the break from close range after collecting another Silva pass.
He rounded off his treble just after half-time, turning home Samir Nasri's cut-back.
Silva completed the scoring in the 65th minute after Tevez's shot had been parried by Steele.
"We scored quickly and scored three in the first half. After that it was finished," Mancini told ESPN.
At Goodison Park, Everton, five-times winners of England's premier cup competition and runners-up in 2009, were booed off by their disbelieving fans as they trailed 3-0 at the interval.
Wigan manager Roberto Martinez told ITV: "I think we deserve a little bit of credit, we stopped them from being dangerous. We knew when to keep the ball, when to go forward, and we looked a real threat every time we got into the final third."
Wigan's appearance in the semi-final at Wembley will be poignant for their chairman, Dave Whelan.
Whelan was 23 when he broke his leg at the famous stadium while playing for Blackburn in the 1960 FA Cup final against Wolverhampton Wanderers. Wolves won 3-0 and the injury ended Whelan's top-flight playing career.
Now 76, Whelan was on his feet in celebration as his team waltzed through their first cup quarter-final since 1987.
Everton were so poor in the first half it was hard to believe they stand sixth in the Premier League table.
David Moyes was hoping to mark nearly 11 years in charge at the club with a third FA Cup semi-final in
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