The veteran Italian left his position by mutual consent after back-to-back defeats to Sweden and Austria effectively ended Ireland's hopes of reaching next year's World Cup finals in Brazil.
Trapattoni, 74, spent five years in charge of the Irish national team, guiding them to the Euro 2012 finals and taking them close to World Cup qualification in 2010.
Keane, 33, Ireland's record cap-holder and goalscorer, said in a statement: "I want to acknowledge the tremendous job that Giovanni Trapattoni did for the past five years for the Republic of Ireland national team.
"In my view he has been a credit to the country and has carried himself with dignity as befitting a man who has given so much to world football.
"It is always sad to see people go but we are all aware it is a part of professional sport. I personally want to thank him for what he has done and wish him well."
National team-mate and new Everton recruit James McCarthy expressed his surprise and sadness at Trapattoni's departure.
The 22-year-old said: "I'm gutted to be honest. He was a good manager and I enjoyed working under him. It is a blow to all the players. It has just happened - I didn't expect it to happen.
"If you look at his record, how well he has done for Ireland, he has got us to the Euros and he fell short, just, of France in a (World Cup 2010) qualifier.
"We were in contention to qualify for the (2014) World Cup but unfortunately we slipped up on Friday and last night we conceded a late, late goal.
"It is disappointing, the players enjoyed working under him."
Queens Park Rangers centre-half Richard Dunne, who returned to the starting line-up in defeats to Sweden and in Austria after a 12-month personal injury nightmare, said: "If you look at the squad we have, we have to be realistic in what we do.
"We very rarely put out a team full of Premier League players and if we do, I don't know how many of them play every week, so it's hard.
"We have a small pool of players to pick from and for us to have been in a play-off and then in a finals...
"All right, we didn't cover ourselves in glory this time, but it has been three campaigns where we can say we have given it a really good shot."