Post Made Community Wiki by Ram Chandra Giri
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No team has ever won a World Cup by winning all of their knockout matches on extra time/penalties and Croatia will be the first team to do so if they manage to do it.

Referencing to my answer on this question, "Most extra time games played in a World Cup", most number of games a team has played extra time (consecutive or otherwise) in a single World Cup is 3 times. So, Croatia will be the first team to play four extra time games even if they don't win it.

References used in my linked answer:

  1. FIFA World Cup™ extra time matches (contains record for matches up-to 2006 world cup)

  2. 2010 FIFA World Cup results

  3. 2014 FIFA World Cup results

  4. Live Match-blog England vs Croatia FIFA official site