The north Londoners looked set to endured a damaging defeat at the Liberty Stadium only to mount a stunning comeback and stay seven points behind pacesetters Chelsea.
Dele Alli, Heung-min Son and Christian Eriksen were all on target in South Wales as Spurs once again proved they can cope without main striker Harry Kane.
Spurs remain in the race for silverware this term but the fact they have not lifted a top-flight title since 1961 recently led Chelsea boss Antonio Conte to hint that the club has grown accustomed to settling for second best.
But Pochettino has appeared to dismiss such claims by insisting Spurs are in good shape with just eight games left to play this term.
"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," he told the BBC.
"Today was a difficult game and the team showed big character."
Pochettino's squad was tested to its limits against Swansea, having lost Victor Wanyama and Harry Winks to injury in their previous encounter at Burnley.
But the Argentine coach, who has had to cope with a large injury list this season, championed the importance of team spirit after seeing some of his fringe-players step up to the plate against Paul Clement's men.
"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.
"It is crazy how the goals came in the last few minutes but we pushed and we created chances to win," he added.
Meanwhile, Eriksen has admitted Tottenham's hopes of catching Chelsea would have been over if they had succumbed to defeat in South Wales.