The Robins won 2-1 at Ashton Gate after extra-time to seal a 4-2 aggregate success, with Johnson ecstatic that his players had stood up to the test from the in-form Eagles.
"I trust these players and they trust me with their fitness and commitment in the last parts of games," he said.
"How many times have we come back in games late on and it's credit to my medical team, my coaching staff and myself of course and the players for working so hard for these types of days.
"It is why you work hard all season because you need to finish strong on occasions like this.
"We won both games and that's put us through to a Wembley final in what is one of the biggest games of the world."