In a thrilling contest, Sterling hit the back of the net twice, putting the home side ahead after his shot looped over the diving Danny Ward and slotting in a cross from Reece James to double their lead.
Chelsea manager Tuchel, who was banned from the touchline on Saturday, said after that he had a "good view" of the match but emphasised there is "no need for repetition" when it comes to watching from the stands.
Speaking on Gallagher's red card, Tuchel added: "You can't make the [second] challenge if you have a yellow card after 20 or 25 minutes. The challenge was a clear yellow card and gave us a huge disadvantage. There was no reason to make this decision [from Gallagher]."
"In the last 15 minutes once or twice we were lucky to not concede an equaliser."
Leicester, who had a goal disallowed in the first half, cut the deficit in half when Barnes found himself on the end of a pass from Jamie Vardy in the 66th minute.
Leicester pushed for an equaliser as Mendy was forced into saves from Barnes and Vardy, who also shot into the side netting.
Leicester boss Brendan Rodgers said he felt that there were many good moments, but his side could have been better at the start of the second half.
"We just started the second half too passively, we weren't aggressive enough. When you go against a side that goes down to 10 men it can be difficult so to go 2-0 down was a real challenge," Rodgers said.
"To be fair to the players they changed the dynamic and we were unlucky not to draw but to lose is disappointing.
"Youri Tielemans has made a couple of great balls, Jamie [Vardy] his timing was excellent and he would normally put the chances away, but he was getting the chances. Harvey Barnes was outstanding for us, we knocked on the door a lot, there was a lot of good for us but we're disappointed with the result.
"We have to press the first goal and have that ruthless streak when defending and if we can do that we will pick up better results."
Chelsea, marshalled in 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 remain in 19th place, with one point.
Information from Reuters contributed to this report.