The Brazilian pulled his side level just before half-time with an incredible free-kick from fully 30-yards which evaded Petr Cech and swept into the top left corner.
But Coutinho had further say in the game, later involved in the build-up for Adam Lallana's effort to put the Reds ahead before volleying in from Nathaniel Clyne's driven cross to make it three on 56 minutes.
The Gunners fought back, having earlier led through Theo Walcott, thanks to Alex Oxlade-Chamberlain and Calum Chambers' efforts, though Liverpool held on at the Emirates Stadium for a vital first three points of the 2016-17 campaign.
Speaking after to the Sky Sports cameras, attacking midfielder Coutinho insisted that the results gives the squad "confidence", not "only" for him but for "the whole team" to come to a top-four rival and win.
"We played a difficult game and played really well," declared the Reds' No.10, who insisted he and his teammates now "have to focus on the next game", keeping taking their development "step-by-step" and "keep going."
Coutinho admits Liverpool weren't at their best to start
It wasn't always so easy for Liverpool, who started poorly and were second-best to a dominant Arsene Wenger side for much of the first-half before Coutinho's equaliser.
The 24-year-old admitted that "in the first 20 minutes" the visitors "didn't play very well" and didn't perform "like the manager wanted", but Coutinho added: "Then, we showed we are fit, we were training hard in pre-season, so we had a good second-half."
Asked about his sensational strike - his first ever Premier League free-kick and an early Goal of the Season contender - Coutinho smiled and said that it was his "best one" but that he will "keep trying" to improve.
He reiterated that it was "a good game" albeit "a tough game" and insisted Liverpool come away "happy because we won" but that they now "have to focus on the next game."
At one stage, Coutinho looked as though he was on for a hat-trick when he seemed set to race through clear on goal, but after misreading the run of Cech in goal - anticipating him to come off his line and meet the ball first - he suffered from cramp.
Shortly after, he was forced off in pain with 20 minutes remaining, although he appeared to improve after initially appearing to have pulled a muscle.
Coutinho's manager Klopp allayed fears of any serious injury in his pre-match press conference when asked about the playmaker's problem, insisting: "[it was] Cramp. Just cramp."