Honduras international Izaguirre - who will represent his country at this summer's World Cup in Brazil - signed for the Parkhead outfit from Motagua in his homeland in August 2010.
The 28-year-old, named Scotland's player of the year following his debut season, has penned a new agreement tying him to the club until 2017, while Swedish cap Lustig has also agreed a similar deal.
The former Rosenberg full-back moved to Parkhead in January 2012 and he told the official Celtic website: "I am absolutely delighted to sign this new contract with the club. 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.
"It is, of course, disappointing that Neil has left the Club as manager and obviously all the players wish him all the best for the future. However, I knew that I wanted to be part of Celtic going forward and I am so pleased that I have been able to commit my future to such a great Club.
"Like all the other players at Celtic, I will work hard with the new manager and do all I can to bring our fans more and more success."
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.
"I would like to thank the Celtic fans for all the support they have given me over these years and I now want to look forward to many more happy and successful times with these supporters."