Arsenal took the lead in the 25th minute when Alexandre Lacazette pounced on a poor clearance by Andrew Robertson and beat Alisson to make it 1-0.
Robertson made amends for his earlier mistake with a goal to put Liverpool ahead at the break, settling a lovely cross from Trent Alexander-Arnold with his chest and clipping the ball past Leno for the 2-1 lead.
Lacazette saw a golden opportunity to draw level just past the hour mark, but Alisson stood tall at the top of his penalty area to deny the Frenchman after he'd broken in on goal.
With full-time approaching, new arrival Diogo Jota marked his Premier League debut with a classy finish, controlling a short Arsenal clearance at the top of the area and nailing a low volley into the net to secure the three points.
"[We were] dominant against a team in form and careful as hell that we did not get caught on the counter-attack," said Liverpool manager Jurgen Klopp.
"Alisson had to make one save, they had two chances in behind but apart from that the football we played was exceptional tonight."
The result puts Liverpool level on nine points with pacesetters Everton and Leicester City atop the Premier League table, while Arsenal, who won their first two games, have now gone 28 games without a win away from home against "Big Six" opposition.
Nonetheless, Gunners manager Mikel Arteta took encouragement from the result.
"We were in the game for a big part of the game. The reality is they are superior to us in many aspects. You could see that in some phases," said the Spaniard.
"I'm really happy in the way the team competed and kept believing. This is the standard we have to reach. We are in a different journey to Liverpool. They've been together for five years and they weren't able to do what they've done tonight two or three years ago, that's for sure.
"They have a lot of good work, a lot of belief in the way they play, great recruitment and they've improved the players that they have. This is what we have to try to do.
"They set the bar and they're probably the best standard that the Premier League has seen in many years. This is where we have to get and obviously we're still a long way from getting there."
The two sides meet again on Thursday in the fourth round of the Carabao Cup (stream live on ESPN+ at 2:45 p.m. ET in the U.S.).