The Gunners needed to win the game by two goals and ended up recording a 3-0 victory.
Ozil played a key role in Arsenal’s opening goal by providing the pass which unlocked Olympiacos defence allowing Ramsey to provide the assist for Giroud.
The French international netted his first hat-trick for Arsenal bringing his goal tally of the season up to 13.
Arsenal take on Aston Villa in the Premier League this weekend. The North London based side sit in second place with 30 points in 15 games.
Olympiakos 3 – 0 Arsenal
Olympiakos: Roberto; Elabdellaoui, Da Costa, Siovas, Masuaku; Milivojević, Kasami; Sebá, Fortounis, Pardo; Ideye.
Manager: Marco Silva
Arsenal: Čech; Bellerín, Mertesacker, Koscielny, Monreal; Flamini, Ramsey; Campbell, Özil, Walcott; Giroud.
Manager: Arsene Wenger