The former Fulham No1 made his Champions League debut in victory over Steaua Bucharest.
"It's exciting," said Schwarzer. "I've always been sat watching these draws from afar so I feel very privileged to be in this position. I want to make the most of it. I'm loving the journey thus far and I want to continue on the journey.
"Demand is huge at this club to be successful and I want to enjoy that aspect of it. I'm enjoying the expectation of winning every single game which is as footballers what you want to do. I want to win trophies; that's why I've come to Chelsea, the same as every other player here and every supporter of this club.
"The inconsistencies have been frustrating for all of us from the top all the way down to the bottom. It's not through lack of effort and lack of commitment but sometimes things just don't quite work.
"If you look at the inconsistency we've had so far - and we're still sitting third in the table and still in touch with Arsenal at the top - it bodes well for when we actually do hit our straps and I would say beware to the rest of the teams in the league."
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