One Burnley fan was stabbed, and four others were injured in the lead up to the match.
The fan in question received medical attention at the stadium, and needed stitches for the injury.
Of around 1000 Burnley fans who made the trip to Greece, 23 were held for being drunk and disorderly.
The Premier League side lost the game 3-1, with the second leg set for Thursday.
In a statement, the club said: "The vast majority [of our fans] were once again impeccably behaved, with the coaching operation to transport supporters to the ground put in place by the club again a great success, as it was in the previous round in Istanbul."
"However, five of our supporters were unfortunately injured - including a person who received a stab wound to the leg - and received medical attention at the stadium."
"Those injured all travelled independently to the stadium, outside the organised travel operation."
The club are investigating the incident alongside Olympiakos and local police.
Burnley have not been in a European competition since 1966/67.
They set up the Olympiakos tie after beating Turkish side Istanbul Basaksehir and Scottish Prem runners-up Aberdeen in extra time.