It was a recovery match for the Blues who were thrashed by Leeds United last week in a 3-0 romping in Yorkshire.
Two yellow cards for Gallagher less than 30 minutes into his first league start at Stamford Bridge seemed to have left Chelsea with a mountain to climb.
But Sterling's double put the hosts in control - the opening goal a deflected, looping shot over goalkeeper Danny Ward in the 47th minute followed by a tap-in 16 minutes later from a Reece James cross.
They were Sterling's first goals since moving to London in the close season from Manchester City and he almost got another when he hit the post during Chelsea's early-second half onslaught.
But Leicester struck back in the 66th minute when Harvey Barnes exchanged passes with Jamie Vardy and drove an angled shot past keeper Edouard Mendy at his near post.
Leicester, who have struggled so far this season, pushed for an equaliser and Mendy was forced into saves from Barnes and Vardy, who also shot into the side netting. Substitute Ayoze Perez hit the bar late on.
Chelsea, marshalled in central defence by Thiago Silva and Trevoh Chalobah, held on for their first home win of the season, leaving them in sixth place with seven points from four games. Leicester are languishing second-bottom with one point.
Tuchel said some of the blame on Saturday also lay with other players who allowed Leicester to burst forward after a Chelsea corner, putting Gallagher under pressure to foul Barnes for his second booking.
"This is not good enough how we take set-pieces," he said. "We have to improve immediately."
Leicester were left to rue a decision by referee Paul Tierney who decided that Mendy had been fouled before defender Daniel Amartey put the ball in the net with a header in the first half with the score at 0-0.
"We are disappointed with the goal that was disallowed," Leicester coach Brendan Rodgers said. "The keeper has made a mistake with his timing... We created opportunities and there's no doubt we should have taken at least a point from the game."
Both coaches declined to comment on reports that Leicester had agreed to sell defender Wesley Fofana to Chelsea. The 21-year-old was not in Leicester's squad on Saturday.