Spurs trailed after 10 minutes thanks to a strike from Wayne Routledge but scored three times in six minutes to secure three priceless points at the death. Chelsea's win over Manchester City ensures a seven-point gap remains between the two rivals but a stirring comeback means there is still plenty of belief emanating from White Hart Lane.
The result was made all the more impressive given the first-team players who were unavailable for the north Londoners, with goalkeeper Hugo Lloris ruled out ahead of kick-off due to illness.
While Pochettino was satisfied with his side's first-half performance, he admitted falling a goal behind meant his side had to find another level.
"We started the game well and created chances in the first few minutes," the Spurs boss told BBC Sport post-match. "We maybe feel that the game is going to be easy. They started to play in a low tempo. When we conceded we realised we needed to improve our level.
"It is crazy how the goals came in the last few minutes but we pushed and we created chances to win. Today was a difficult game and the team showed big character. The most important thing is the badge. When you play for Tottenham it is not about the names, it is about the team. This season we are showing that we are a team.
"I don't care what people say or what people think of the history of the club. This season we are fighting again. We are in a good way."
Tottenham, who were also missing injured duo Danny Rose and Harry Kane, toiled in the first-half and found themselves a goal behind early on.
The introduction of Vincent Janssen in place of the disappointing Moussa Sissoko in the second half aided the fight back as they cranked up the pressure on Swansea, with Dele Alli scrambling home the equaliser with two minutes of normal time remaining.
A break in play which saw Swansea goalkeeper Lukasz Fabianski require lengthy medical treatment meant there were seven minutes of added time – more than enough for Spurs to turn the game on its head.
Janssen played a pivotal role in the second, receiving the ball from Christian Eriksen before superbly flicking it around his marker into the path of Son Heung-min, who found the back of the net via a deflection.
With the home side pushing for an equaliser, Tottenham picked them off on the break with Eriksen claiming a deserved goal in the fourth minute of added time.