The Gunners were grateful referee Anthony Taylor reversed his decision to award Burnley a second-half penalty as they followed up last week’s first league win of the season against Norwich with another three points.
Arteta said: “It was a really hard-fought victory in a place that is really tough to come and win. You have to be ready for a fight.
“They do really well what they do. We started really well, we had some big phases where we saw a lot of things I wanted as a team.
“[Burnley] split the two blocks very much and then you have to defend crosses and long balls and we did that really well.
“The commitment and the attitude that my players showed throughout the game is the most pleasing thing for me.”
Odegaard’s superb 25-yard free-kick, after Ashley Barnes had tripped Bukayo Saka, proved decisive.
Burnley were aggrieved at referee Taylor’s U-turn after substitute Matej Vydra had gone down in the box under goalkeeper Aaron Ramsdale’s challenge, but Arteta insisted VAR intervention resulted in the correct decision.
The Spaniard said: “The assistant was saying straight away it’s not a penalty. You have to overturn that.
“Hopefully this can build confidence and trust and belief that we can win in difficult matches. We need more consistency on that.”
Burnley’s winless home Premier League run was extended to 13 matches in total and they have taken only one point from their first five matches this season.
Dyche and his players were left furious by the penalty decision, but later accepted Ramsdale had made contact with the ball before clipping Vydra.
The Burnley boss said: “It’s not a major surprise with our penalty record. I must say the keeper does get a touch on it, I’ve seen it back, so in theory that doesn’t get given.
“You might question why [VAR intervened], if it’s not a clear and obvious error, but it obviously was to go to the VAR screen.
“Last season [against Leeds] the one on [Patrick] Bamford gets given, this season that doesn’t, but that’s sometimes the way it goes.”
Dyche, who signed a new four-year deal earlier in the week, handed summer signing Maxwel Cornet his debut off the bench.
The Ivory Coast international made an instant impact, forcing Ramsdale into a flying save within minutes.
Dyche added: “He did very well with the ball, he has to learn the little nuances of the Premier League and transition, but we brought him here because we feel he is a good player.”
Rival fans clashed briefly in the Barnfield Construction Stand at the final whistle, hurling plastic bottles, prompting repeated calls over the tannoy for them to leave the stadium.
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