The German scored as a first-half substitute against Paris Saint-Germain on Tuesday, before Demba Ba struck three minutes from time to see the Blues cancel out a 3-1 first-leg deficit to advance on away goals.
Chelsea twice hit the woodwork in the second period as it appeared that the French side may hang on at Stamford Bridge, but were finally rewarded for leaving "everything" that they had on the field, according to Schurrle, when Ba netted late on.
"We needed to have patience, we hit the bar twice but we never gave up, that's what the team is about," he told Sky Sports News.
"When you see on the clock there is five minutes to go and you need to score, you don't give up. You put everything you have in the game."
Chelsea manager Jose Mourinho has now reached the last four of the competition a record eight times.