Cesc Fabregas' 50th Premier League goal earned the Blues a slender 1-0 win over Swansea City at the Liberty Stadium on Saturday, which left Antonio Conte's side two points adrift of Tottenham, who have a game in hand.
Mauricio Pochettino's side host Watford on Monday and are in pole position to secure fourth, but Cahill wants the Blues to be ready to pounce should there be further slip-ups by their London rivals.
He told Match of the Day: "It's a massive three points for us. We've got to try and win every game until the rest of the season.
"The teams above us don't slip up and give up many points but let's keep winning and see what they do."
Fabregas, whose fourth-minute goal was a superb whipped effort into the top corner, echoed Cahill's sentiments.
"Every game we have to show our pride and show we want to make it happen, put pressure on the teams above us," he said.
"In football, anything can happen and we have to make sure we take it if it comes."