The former Liverpool and Manchester City star has struggled to show his best form with the Blues.
Sterling was the subject of boos from his own fans after he missed a penalty earlier in the game.
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After the match, Sterling wrote on social media: Wembley Bound. To all the Chelsea fans, apologies for the penalty miss.
I'll be back 10x times stronger to help the team win and continue to keep fighting for the badge every single day. Let's go Blues."
Cole can miss, Raheem can miss, look for me, the decision is there and of course I'm going to support always, the decision of my players on the pitch," manager Mauricio Pochettino said on Sterling taking the spot kick instead of Cole Palmer.
That is about to grow and be more mature and of course, always happy we take decisions."