The Merseysiders were stunned 3-0 by the unfancied Latics in their quarter-final at Goodison Park and City were anxious to avoid an even greater upset against Barnsley later the same day.
The Premier League champions responded by thrashing the Championship strugglers 5-0 at the Etihad Stadium and will now face rivals Manchester United or Chelsea for a place in the final.
Lescott, a former Everton player, said: "We knew it would be tough, but we had all watched the game earlier.
"Everyone predicted an Everton win, but Wigan surprised most people.
"We didn't want that to happen to us so we had to put in a professional performance.
"The manager set our stall out before we even travelled to the stadium, saying that if we were professional and played in the right way it would be hard for Barnsley to create many chances.
"From the get-go we proved that, and made it hard."