Tevez, who controversially joined City last year in a move that has heightened the rivalry between the two clubs, wiped out United's early lead when he smashed home a 42nd-minute penalty.
The Argentina forward headed their second goal against the holders after 65 minutes when he was left unmarked in United's six-yard box with only goalkeeper Edwin Van der Sar to beat.
United went ahead after 17 minutes when City failed to deal with an Antonio Valencia cross leaving Ryan Giggs with a simple tap-in after City keeper Shay Given blocked a Wayne Rooney shot.
The first semi-final between the two Manchester clubs since 1969 will be decided in the second leg at Old Trafford next Wednesday with the winners meeting Aston Villa or Blackburn Rovers in the final at Wembley Stadium on February 28.
Villa hold a 1-0 lead after the first leg at Blackburn.
United settled first at a sold-out Eastlands and took the lead after Patrice Evra picked out Valencia on the right. His low cross into the box found Wayne Rooney and although Given saved his shot the ball ran to Giggs, a former City youth player, who scored his first derby goal since 1996.
United dominated much of the half after that but City, whose Italian manager Roberto Mancini was experiencing his first Manchester derby, equalised after a controversial call by referee Mike Dean.
United defender Rafael tugged at Craig Bellamy's shirt outside the penalty area and, while the City forward tried to stay on his feet, he fell inside the box.
Dean ruled there was another foul inside the area but TV replays clearly showed there was not.
Tevez ignored the pleas of his former team-mates and the time-delaying tactics of Van der Sar, which earned the keeper a booking, to blast the penalty into the net.
With United personnel claiming in recent days that the club were right to let Tevez leave at the end of last season, the Argentine cupped his ear with his hand to celebrate - and teased the United bench again after his second goal.
That came when he made it 2-1 with his 13th goal in his last 11 games, taking advantage of some slack United defending to head in from close range.
United attacked relentlessly for the equaliser for the remainder of the match with Rooney twice being thwarted by Given and substitute Michael Owen having a shot saved on the line.
Although the visitors failed to find another goal, the tie is still evenly balanced with next week's second leg likely to be just as lively.