The defending champions are back in control after Liverpool drew 0-0 with Everton in the Merseyside derby on Sunday, giving City a one-point lead at the top.
With nine games to go, Jurgen Klopp's men must chase City, who beat Bournemouth 1-0 on Saturday.
Robertson feels it is Pep Guardiola's side under pressure, even though Liverpool are aiming for a first league title since 1990.
"We become the chasers now and all the pressure is on Manchester City," he told the BBC.
"We will fight for every last ball until the whistle is blown at the end of the season. The pressure has been on City because they are expected to retain the title.
"Everton think they have stopped us from winning the league. It shows how well we are doing that they want to stop us."
Following Sunday's setback, Klopp also stated that he has not given up on the title race and that his players have not either.
The Liverpool boss also added that he has faith in Mohamed Salah despite recent struggles as the Egyptian star has not scored in three consecutive matches for the first time since joining the Reds prior to last season.
Liverpool have won just two of their past five league games and are next in action at home to Burnley on Sunday.
The two sides settled for a 0-0 draw in the opening leg at Anfield, leaving it all to play for in their round of 16 second leg clash.