The sides made the free-agent deal Friday.
Ross often caught Lester in Boston during the past two seasons, including both of the left-hander's starts in the 2013 World Series victory. Ross started behind the plate for Lester's last 11 starts this year before the Red Sox traded the pitcher to Oakland.
Ross, who will turn 38 in spring training, hit .184 with seven home runs and 15 RBIs in 50 games for Boston last season.
The Cubs got two-time All-Star catcher Miguel Montero in a trade with Arizona this month. Chicago also claimed former Boston catcher Ryan Lavarnway off waivers from the Los Angeles Dodgers on Friday, and the Cubs still have Welington Castillo behind the plate as well.