The former Germany international returned to the Gunners first team against the Clarets and played a role in two of the hosts' three goals.
Unai Emery also entrusted Ozil with the captain's armband for the fifth time in the top flight this season andafter the game insisted the 30-year-old had a long-term future at The Emirates Stadium .
And Kolasinac says Ozil has had the full backing of the squad during the period of uncertainty.
The Bosnian defender told Sky Sports : "That's the case when you're a big star, like Mesut is.
"Of course there are always negative headlines but, as a team, we don't pay attention to them and Mesut doesn't either.
"That's what's most important. We're all behind Mesut - all the staff, all the players.
"If negative things are written, then that's just what happens. In life, not everything is positive - sometimes there are negatives written about you too, and he's had that.
"But with his performance [against Burnley] he's quietened a few critics."
Kolasinac played a role in both of Pierre-Emerick Aubameyang's goal against Sean Dyche's men, as Arsenal returned to winning ways after back-to-back defeats.
The opening goal came via a wondrous Ozil pass which found Kolasinac in an advanced position, before Aubameyang converted.
The ex-Germany youth international said of the pass: "It was unbelievable.
"We all know Mesut's qualities and I only actually made that run because I saw that he had the ball. Not many players would have been able to play that pass.
"He can, and when I saw him get on the ball, I knew he was going to find me. It was a brilliant pass from him."
Emery has shared the captaincy between a clutch of senior players with Petr Cech, Granit Xhaka and Aaron Ramsey given the armband this term.
And while each player has their own leadership style, Kolasinac has revealed Ozil has a passive approach.
"Mesut's not the sort of player who will scream at you in public just to show the people on the outside that he tries to push us on," he added.
"He'll speak in the dressing room, he'll go to every player individually. With his quality, he brings a lot to this team. He's very important for us as a captain and as a player."