The goalkeeper has penned a three-year contract.
Pope admits he is lucky to have been given the opportunity after Tom Heaton dislocated his shoulder.
“It's been lucky on my part how it's come about – a bit by default. But I've really enjoyed it since I've been in the team," Pope told the Burnley website.
“It's been a good few weeks for me personally as well. Everyone at the club is happy we've made a good start and we've got to build on that now."