The Turkish outfit advanced from the last-16 stage on Tuesday after knocking out Schalke 04 4-3 on aggregate following a 3-2 victory in Germany.
Yilmaz, who is the tournament's joint-top scorer with eight goals, wants Galatasaray to progress from their pending quarter-final match and reach the final four of Europe's elite competition.
"It is a very nice night," said Yilmaz. "Everything went like we wanted and we believed in our triumph. I would like to thank the Turkish nation. Our hearts are with them.
"Now everybody is celebrating and dancing in the dressing room. Our new target is reaching the semi-finals.
"There will be really good teams in the next round but it will not matter who we are paired with. We can go on the pitch and fight like lions. We have already surpassed expectations."
Yilmaz struck the side's second goal of the night on Tuesday.