Sterling scored his first goals for Chelsea following his £50million switch from Manchester City in July and Blues boss Tuchel was delighted to see the England forward "step up" after Conor Gallagher's dismissal.
Tuchel said of Sterling's contribution: "It was needed. We need him to score, it's what he does and he will score.
"I could feel he wasn't happy, he wants to score more and have more chances. So we played in a more aggressive shape in the first half for 25 minutes and then we are a man down, so we needed him to step up.
"He did, and he did what he always does. He will score, I'm convinced, and he will create chances.
"The goals were crucial today because they gave us belief and the boost that it was possible. They were responsible for the win."
Despite the win, Tuchel was unimpressed with his side's third red card in three games.
The German was serving his touchline ban for his red card in the draw against Tottenham, while Gallagher received his second yellow card - both for fouls on Harvey Barnes - in the 28th minute. Chelsea also had Kalidou Koulibaly sent off against Leeds.
Tuchel added: "Don't include mine, it was after the final whistle, and the other ones, to be precise, were two yellows, yellow-red.
"Two of them totally were unnecessary and that is part of the analysing of the situation.
"Twice it is absolutely not necessary and to give such a huge disadvantage we have to stop doing this.
"We cannot behave like this if we are on yellow cards. We have to learn very, very fast. This must be the absolute exception from the rule and cannot be repeated."