The result left the champions nine points behind leaders Manchester United but Hart maintains the City players remain confident.
The England goalkeeper said: "We have got to be. There is no point turning up otherwise. We have got to win every game and look not to drop any points. It is a good challenge for us.
"We will always be in it, but unfortunately it's a big gap, getting too big for our liking.
"You have to move forward, and that means we have to beat Southampton next week. We are nine points behind with 13 games to go, so we need the win."
City's defence had gone 584 minutes without conceding when Daniel Sturridge scored Liverpool's opening goal against his former club.
Hart added: "We need to do what we have done in the last seven games, when we won six and drew one.
"Of course we have now drawn again, and that is enough of the draws. We need to start winning."
Hart, 25, added: "It was a tough game. I thought Liverpool played really well. I imagine they'll argue we were lucky to get a result in the end. That's a shame but you have got to move forward.
"They are a good side and I think they have finally worked out what the manager wants from them. They are a really tough side to play against.
"It is not the result we wanted but I think a draw is a fair result."