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.
"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."
Ireland might have won far more comfortably had it not been for the efforts of Nielsen, who produced a string of important saves as his defence came under sustained pressure.
However, Trapattoni was satisfied with a victory which at least keeps alive his team's hopes of finishing second in the group behind runaway leaders Germany.
Trapattoni said: "Obviously this evening, it was very, very important to win the game and we won.
"People want five, six, seven, eight goals - it's not so easy, but we played well, particularly in the first half. We had three or four or five opportunities to score.
"The players said to me they had the sun in their eyes and they lost the moment to pass the ball, but after, we deserved to score."
Austria's win means they are now leading the race for the runners-up spot, although the Swedes host the Faroes in Solna on Tuesday evening.
Asked if he was surprised by the result in Vienna, former Red Bull Salzburg boss Trapattoni said: "No, because Austria are a good team. I was in Austria and I know many players."
"Without a silly mistake, we could have won [against Austria in March], but they are a good team and they have good players.
"But we are at the same level. Without this silly 90th-minute goal, we deserved to win and we could have been in second place behind the Germans."
Trapattoni and his players head for New York tomorrow for Tuesday's friendly against Spain, although Shane Long and McGeady will not be travelling because both are to be married, while Marc Wilson is due to have surgery to remove a metal plate from his leg.