Tottenham had to be patient, with the opening goal coming after 66 tight minutes at Turf Moor, and Pochettino hailed the importance of a win that was just Spurs' third in their last 11 away matches in the league.
"I'm very proud because it was a really tough game for us," Pochettino told BBC Sport. "We showed great character and I'm very proud and pleased because the performance in the second half was very good.
"This win is massive for us - we have to be there if Chelsea fail and we are there. We are fighting for the Premier League.
"When you reduce the gap to seven points it's completely different to 10 points - we just have to be there if Chelsea fail and want to be there until the end of the season."
But injuries marred the win for Spurs as Victor Wanyama and Harry Winks both had to be replaced shortly before half-time. In a freak incident, Winks was late into a challenge on Stephen Ward and momentum sent the young midfielder crashing into the ground right in front of the Burnley dugout and he was given oxygen on the sidelines.
Pochettino said more time is needed to assess Winks' ankle injury, although Harry Kane's recovery is progressing well.
"We're still waiting on Harry Winks but Victor Wanyama got a big knock on his back and hopefully it's not a big issue for him," Pochettino said. "Harry Kane is good, he's doing things on the pitch and may be back in a few weeks."
Defeat saw Burnley slip to 15th after their winless run stretched to seven matches but Sean Dyche was unconcerned by his side's cushion from the bottom three being cut to five points.
"I thought for the first half it was a very good performance," Dyche told BBC Sport. "And in the second half we started strong and we were in it, but we gave a really poor goal away and it changed the whole feeling.
"They're a good side obviously, they're second in the Premier League for a reason and they kept the ball better. I was pleased with the team we picked and the way we delivered a performance until the goal, which defused us and it gave them a lift.
"We knew this was a tough part of the season and we're still in good shape. We've earned the right to be in good shape and we need to focus and concentrate on the challenge in front of us. The next game [at home to Stoke City on Tuesday] comes quickly."