The Greek league leaders started the match at Georgios Karaiskakis Stadium in up tempo style and scored in 36th minute when veteran midfielder Esteban Cambiasso hit a bullet header from a well-placed cross by forward Karim Ansarifard, beating Besiktas' goalkeeper Fabri.
The Turkish league leaders stepped up their game in the second half and created more chances. Their hard work was rewarded in the 53rd minute when striker Vincent Aboubakar found the net after a terrible clearance from Olympiacos goalkeeper Nicola Leali.
Besiktas increased the tempo after that, however, it was unable to pull ahead as the match ended with nothing between the two sides.
"We knew beforehand it would be a tough game. We were well prepared. It was important to score the away goal, but nothing is decided yet,” Man of the Match Vincent Aboubakar was quoted by UEFA's official website.
"We had balance in our game, we kept the lines compact and that is a ground for optimism. Our coach gave us some pointers at half-time and I believe we followed them in the second half," he added.
The second leg of the tie will be played in Istanbul’s Vodafone Arena on March 16.
Here are other results in the first leg of Round 16:
Copenhagen - Ajax: 2-1
Rostov - Manchester United: 1-1
APOEL Nicosia - Anderlecht: 0-1
Gent - Genk: 2-5
Olympiacos - Besiktas: 1-1
Lyon - Roma: 4-2
Schalke - Monchengladbach: 1-1
Celta Vigo - Krasnodar: 2-1