Kostas Fortounis scored twice for the hosts - his second from the penalty sport after Ben Gibson was sent off for a second yellow card when handling the ball.
Andreas Bouchalakis also netted for Olympiakos, with Chris Wood’s penalty giving Burnley hope of a comeback in the second leg next week.
And manager Sean Dyche was unhappy with Slovenian official Slavko Vincic’s decision to award the penalty for Gibson’s handball.
Dyche felt Vincic’s attitude switched after Olympiakos players mobbed him at half time, complaining about not getting decisions in their favour during the first 45 minutes.
A Burnley fan was stabbed before Thursday's Olympiakos game in the Europa League (Image: REUTERS)
Around 1,000 Burnley fans made the journey to Athens (Image: REUTERS)
Burnley fan went down to 10 men when Ben Gibson was sent off (Image: REUTERS)
Invalid email
We use your sign-up to provide content in ways you've consented to and to improve our understanding of you. This may include adverts from us and 3rd parties based on our understanding. You can unsubscribe at any time. Read our Privacy Policy
Olympiakos and Burnley are battling to reach the Europa League group stage (Image: REUTERS)
Chris Wood scored a crucial away goal for Burnley (Image: REUTERS)
“The scenes I saw at half-time from all and sundry… and then things change in the second half,” said the Burnley boss.
“They were waiting for the referee. You all saw the feel of the game was different in the second half.
“They miraculously get a penalty - when one of our player’s blocks - it’s his hip and then hand and that’s deemed a professional handball. We’re all left scratching our heads.
“We are told to respect referees. I feel for our fans, they’ve come a long way to see a balanced game and haven’t seen that.”
Olympiakos will take a 3-1 lead into the second leg (Image: REUTERS)
Burnley had enjoyed a mixed start to the season before Thursday’s game.
They have already knocked out Aberdeen and Istanbul Basaksehir in Europa League qualifying, but one point from two Premier League games has seen then get off to a worrying start on the domestic front.
The Clarets travel to Fulham on Sunday before hosting Olympiakos in the second leg of their Europa League playoff next Thursday.