The 32-year-old LA Galaxy striker plundered all the goals in a 3-0 win on the night he became Ireland's most-capped player, enabling his side to climb into third place in Group C following Sweden's 2-1 defeat in Austria.
Trapattoni said: "I particularly wish to congratulate our captain because for 30 years, I have known the Irish team and the Irish players and for me, he is the best Irish player I have known in 30 years."
He added: "I wish to congratulate him because he scored. At the moment, he is in good form, but I repeat what I said yesterday, he is also one of the best players in Europe."
Keane needed just five minutes to celebrate his 126th senior international appearance in style when he swept the ball past goalkeeper Gunnar Nielsen from Aiden McGeady's cross.
Despite coming under concerted pressure, the Faroes held out until half-time, but Keane was in the right place at the right time once again 10 minutes after the restart to convert Seamus Coleman's inviting ball.
Keane capped the perfect evening from close range with nine minutes remaining when substitutes James McClean and Conor Sammon joined forces to present him with a tap-in, his 59th international goal.
Keane told Sky Sports 1: "All in all, it's been a great occasion for everybody. I'm glad it's over now. There was a lot of fuss before the game, but the important thing obviously was concentrating on playing football. We did that and got the three points, so everybody's happy.
"There was a lot of talk about getting the record caps, but for me that was almost in the back of my mind. It was about playing the game and playing in the right manner and getting the right result for the team.
"Certainly it was a special occasion for everybody involved, my family, but I was just concentrating on the game and getting the result for the lads."
- Sports & Recreation