The Gabon forward gave Arsenal the lead in the 14th minute, knocking in a pass from Sead Kolasinac after a classy ball from Mesut Ozil. He then smashed high into the net in the 48th to complete an impressive counter-attack from Unai Emery's side.
Burnley, 18th in the table and without an away win since September, hit back with a scrappy goal from Ashley Barnes having spurned a good chance to take an unlikely lead at the start of the game when Ashley Westwood was denied by the feet of Arsenal goalkeeper Bernd Leno.
Substitute Alex Iwobi restored Arsenal's two-goal cushion with a simple finish in stoppage-time, helped by another clever move from German playmaker Ozil, who was left out of the team for Wednesday's 2-0 League Cup defeat by Tottenham Hotspur.
Aubameyang's brace moved him on to 12 goals for the season, leapfrogging Liverpool's Mohamed Salah as the league's top scorer and taking fifth-placed Arsenal on to 37 points, level with Chelsea who play Leicester City later.
Arsenal's 22-game unbeaten run in all competitions was halted on Sunday with a 3-2 defeat at Southampton and they were knocked out of the League Cup at the quarter-final stage with a painful home loss to north London neighbours Tottenham.
They were given little respite from a highly physical Burnley side but ultimately had too much quality for Sean Dyche's side.
"We expected it to be tough, we prepared the match thinking about what they would do, that they would be strong in the air, so we made sure we won the duels and we also wanted to impose our game plan," Arsenal coach Emery told reporters.
"We could not play as we wanted to for 90 minutes but we are very happy with the victory. Physicality is part of football, you always need to be strong as well as impose your quality, and our victory today is a consequence of this."
Burnley coach Dyche credited his team for causing Arsenal problems, but cursed their own defensive vulnerabilities.
"Overall I was pleased with the performance, especially the mentality to come here and take on the game," he said.
"A couple of their goals were soft, but attacking-wise we were a problem today for them. We could have got something but we know even with a good performance you can't afford to give away goals like the two we let in."
(Reporting by Richard Martin, editing by Ed Osmond)
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