The defender's goal got his team into the game in the second half, and the England international said that the players worked hard to beat the Saints.
"It was the worst possible start, but we showed character to come back," Cahill told Sky Sports News.
"We needed a result and we showed [our commitment] in the second half. The gaffer (Jose Mourinho) just told us to keep going and not to worry.
"He stressed not to worry and we're pleased with the second-half performance."
The Blues have moved up to second with the win.