Argentina striker Tevez, who joined City for 25.5 million pounds after two seasons at Manchester United on a lease agreement, headed in with 20 minutes to go after Shaun Wright-Phillips had put the visitors ahead after halftime.
A full-strength City side were put under intense pressure by their Championship (second division) opponents in the first half at Selhurst Park with Lescott, who joined from Everton for 22.5 million pounds Wednesday, kept busy.
City, who have begun the season impressively with two victories in the Premier League, moved up a gear in the second half to book a place in the third round of the competition.