Dortmund defeated Borussia Monchengladbach 2-0 at Signal Iduna Park this afternoon to become the first team since Bayern Munich in 2006 to retain the crown.
It was also the first time Bayern have gone two seasons without winning the title since Dortmund triumphed in successive years in 1995 and 1996.
"What the team has done, that's madness," Klopp, whose feats at Dortmund have made him one of the most highly-rated young managers in European football, was quoted as saying by kicker.de.
"There are no words for it. If there was ever a deserving champion, then we are it.
"My team has an extraordinary character. This is crazy what we have done."
After losing three of their opening six Bundesliga games of the season, Dortmund have not been beaten in the league since September.
Second-placed Bayern beat Werder Bremen earlier on Saturday to give Klopp's men the chance to clinch the title in front of their home fans, which they duly did thanks to goals from Ivan Perisic and Shinji Kagawa.
Dortmund chief executive Hans-Joachim Watzke said: "The team played wonderfully today [Saturday]. I am pleased that Bayern won in Bremen, otherwise we would not have experienced this today. A perfect Saturday. To put in such a perfect performance shows the quality of the team.
Dortmund will now meet Bayern in the final of the DFB-Pokal as they look to win the double for the first time.
Germany coach Joachim Low revealed his admiration for the Dortmund side.
"It is impressive that Borussia Dortmund have won the title for a second straight year with this young team," he said.
"My complements go to coach Jurgen Klopp, who has done excellent work. In general what impresses me at Dortmund is the unity at the club. Between coach, team, the board and fans there is a real cohesion."