Goals from Samuel Eto'o and Gary Cahill saw Chelsea claim a comfortable 2-0 victory at Stamford Bridge to oust the Turkish side from the competition 3-1 on aggregate.
The Londoners can make it three European titles in a row - they won the Champions League in 2012 and the Europa League last season - if they reach this year's final and win it, but the likes of Barcelona, Bayern Munich and Real Madrid stand in their way in the last eight.
Veteran midfielder Lampard told ITV: "I won't make any predictions yet. There are number of great teams left in it and we'll wait and see who we get drawn against.
"We're confident at the minute but it's very tough and gets tougher every year. But we're still in it."
It was a straightforward evening for Chelsea, who took the lead through Eto'o inside five minutes before Cahill's close-range finish left Galatasaray chasing shadows. Returning striker Didier Drogba was limited to a few half-chances.
"It was everything we wanted it to be," Lampard added. "It was a tough game and we didn't take for granted that we would get a decent result out there or think we were going to win comfortably.
"They hardly threatened us apart from right at the end through Didier and I think it was a very controlled and very professional performance from all of us.
"Everyone knows what Didier's about, he's top class. But it's not just Didier, they've got threats in different positions."
Despite the achievements of his predecessor Rafael Benitez, current Blues manager Jose Mourinho has spoken of his disregard for the Europa League and is relieved Chelsea are now back among "the best clubs in the world".
He said: "We are there (in the last eight) - but it's different from last season because last season was the Europa League at a different level.
"To come from the Europa League to the quarter-finals of the Champions League is important for the club, important for the players.
"We are a Champions League club and now we are among the best eight, the best clubs in the world, and we are there.
"The players deserve to be there."
Should Manchester United fail to overcome a 2-0 deficit in Wednesday's second-leg meeting with Olympiacos, Chelsea will be the last English team standing in Europe's elite competition.
Mourinho, however, is willing David Moyes' men to make it through.
"I hope it's not true," he added. "I hope United do it tomorrow."