Sterling, who arrived from Manchester City this summer, somewhat fortuitously bundled the ball home from close range at the second attempt in the 37th minute following a fine through ball from Jorginho.
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By that point, Chelsea were already a goal to good. N'Golo Kante opened the scoring with 20 minutes gone as he was allowed to drift upfield before unleashing a fine left-footed striker into the bottom corner from the edge of the penalty area.
But Chelsea would not be denied a much-needed win ahead of the start of their Premier League season against Everton a week on Saturday. Sterling almost grabbed his second of the night with an effort that came back off the post before Mason Mount turned in substitute Callum Hudson-Odoi's cross in the 90th minute.
Chelsea came into the game following a disappointing 4-0 defeat to London rivals Arsenal and a loss on penalties to MLS side Charlotte FC as the club prepares for their first season under new ownership.