That win ensured Tottenham will remain in second, as Chelsea's nearest challengers - barring a huge win for Manchester City against Liverpool - and Vertonghen is adamant his side remain in the title fight.
"Obviously Chelsea are the favourites at the moment for the title," he told Sky Sports. "But we won't give up until they lift the trophy.
"I think we're doing well. We just have to aim for the highest [position] possible - at this moment, it's second place. We will push for first place.
"We have to keep winning our games. The other teams [in the top six] are dropping points and playing each other, so we have to make sure that gap is big enough to secure [qualification for] the Champions League as soon as possible."
The win against Southampton came without injured star Harry Kane, with Eriksen reflecting that the whole team worked to compensate for his absence.
"Everyone felt the responsibility without Harry," he said. "We have 11 players to replace Harry when he's not here, but the players are good enough to substitute [in for Kane].
"It was very important that someone else scored, of course, and we had to keep it going."
Eriksen did, though, concede that Spurs had to adapt with Son Heung-min playing up front instead of Kane - something he felt they did well before Southampton came back into the game in the second half.
"[The role] was exactly the same," he added. "We still played the same system, it was just a different player, Sonny, up top instead of Harry. [Son] is a dribbler, he wants the ball at his feet to make his runs and he's a different player.
"I think we did really well in the first half, but, obviously, the second half was a bit tougher."