In Jurgen Klopp's final Premier League showdown against Pep Guardiola, Liverpool trailed at half-time to John Stones' opener from Kevin De Bruyne's corner.
But the hosts roared back in the second period after Alexis Mac Allister levelled from the penalty spot.
Despite missing a number of key players to injury, Liverpool missed a host of chances to open up a four-point lead over City, who are aiming to become the first side to ever win four consecutive English top-flight titles.
A share of the spoils leaves Arsenal top on goal difference from Liverpool with City one point back in third.
"We had very good chances, unfortunately we couldn't get the winner but overall bittersweet," Van Dijk told Sky Sports.
"Based on the second half you feel we should have won it, but they hit the post as well. It was so close.
"Manchester City has been very successful with the titles they have won and every game we have played against each other have been very intense. Every game is so difficult, they have quality all over the pitch and that is why they won the treble last year. I respect that but the aim for us today was to win the game.
"To come away with a point is not a bad situation and we have to focus on what's next. We just have to enjoy the ride."
City's only victory at Anfield since 2003 remains a 4-1 win behind closed doors three years ago.
And Guardiola's men face another huge game in the title race in their next league match at home to Arsenal on March 31st.
"We know what we are in for coming here," said Stones.
"We feel a bit deflated in the dressing room. We came here to win, but this is such a difficult place to come.
"We got a point out of the game and this is such a difficult place to come. It's a bit raw for me right now to analyse the game. We know what a top side they are and what they bring and we've had a lot of really good games against them.
"The second half sort of balanced it out today but we'll have to take the positives out of today and move forward and try and go on a nice run."
