Tottenham were staring down the barrel in the third round replay at White Hart Lane as a superb half-volley from Marvin Sordell and Ross Wallace's deflected free-kick put Burnley into a 2-0 lead after eight minutes.
However, Paulinho's expert finish brought the hosts back into the match in the 10th minute and Etienne Capoue drilled home the equaliser shortly before half-time.
Vlad Chiriches punished Burnley goalkeeper Tom Heaton's error to head home Tottenham's third shortly after the restart and Danny Rose scored a rare goal to round off the win and set up a fourth-round clash with Leicester City.
And head coach Pochettino thought his team were well worthy of a place in the next round.
"It was a very interesting game. It was a strange game," said Pochettino.
"We deserved to go through to the next round. It was key to go in at half-time at 2-2.
"We fixed something from the tactical situation and scored two goals quickly to decide the game.
"The team has shown character and faith. It's important to grow in the group.
"I am happy they showed character because that is important."
