The Gunners were made to work hard but scored three second-half goals to record a 3-0 win at the Emirates Stadium, and the ex-Southampton youngster was delighted to get on the scoresheet.
Chambers told Sky Sports News: "That was my first goal and it was a really proud moment for me. I've scored at youth level but never before for the first team, so I'm really pleased.
"It was tough, [Burnley] sat back and defended well, we just had to keep working hard and take our chances."
The versatile England international also reserved a word of praise for Alexis Sanchez, after the Chilean scored the other two goals against the Clarets.
"He's a great player," added Chambers. "He works so hard for the team and creates chances. We're picking up momentum and we've got to keep training hard and take it into the next game."
The Gunners are currently fourth in the Premier League table.