The pair, who have been in talks with Celtic over the last few months, have put pen to paper on three-year deals.
"I am absolutely delighted to sign this new contract with the club," Lustig told the club's official website. "Ever since I came to Celtic I have felt welcomed and I have enjoyed every minute of my time here.
"It has been brilliant to win trophies here and I hope we can go again next season and bring more silverware to Celtic and achieve one of our main objectives in the next few months and take Celtic to the Champions League again."
Izaguirre added: "Celtic is a place I have called home for a number of years now and it is the place where I want to be in the future. It is a magnificent football club in a great city, somewhere my family have settled so well."