Liverpool found themselves level against the run of play going into the second half at the Emirates Stadium, Gabriel's own goal equalising for the Reds after Bukayo Saka had opened the scoring.
However, despite some slight improvements after the restart, Liverpool fell behind with 67 minutes played, Gabriel Martinelli capitalising on a rare error from both captain Van Dijk and goalkeeper Alisson.
Van Dijk attempted to shield a long ball over the top for Alisson to clear, but the two bumped into one another and missed the ball entirely as Martinelli snuck in to fire into an empty net.
Liverpool went on to concede a third late on after Ibrahima Konate's red card, with Leandro Trossard's strike deflecting off Van Dijk on its way through Alisson's legs.
The result saw Liverpool's lead at the top of the table cut to two points, but Jurgen Klopp refused to kick Van Dijk and Alisson when they were down following their blunder.
"I would have loved to see the game longer at 1-1," the boss stated. "We found a way into the game. No shots on target but we scored a goal.
"Growing more into the game, second half, we had our moments and then we concede that goal. That doesn’t help. It just shows the boys are human beings. Today we were really human."
Speaking later, Klopp added: "The story of the second half we will never know how it would’ve been if we don’t concede the goal, but our two main guns have a misunderstanding.
"That just shows, I’ve said it a couple of times, they’re human beings. That actually makes the things they usually do even more special.
"Sometimes you forget that they are humans as well. For today, they are not happy, we are not happy. But that’s it now, I don’t need to make more of an away game at Arsenal."
The Reds will need to recover from a disheartening defeat next Saturday when they welcome Burnley to Anfield, while a clash with Man City in early March now holds even greater significance.