The Arthur Miller classic captured honors for best revival of a play, best leading actor, best director and best featured actress, while Miller, 83, won a special Tony for lifetime achievement.
"Just being around to receive it is a great pleasure in itself," Miller told the audience at Manhattan's Gershwin Theater.
Brian Dennehy won best actor at Sunday night's Tony Awards for his role in Death of a Salesman, while best actress went to Dame Judi Dench for her performance in Amy's View. But it was comedian Martin Short, who won for best leading actor in a musical (Little Me), who managed to steal the show.
"There are so many people that I really must thank and should," said Short in his acceptance speech. "But the reality is: I did it all myself."
The broadcast of the Tony Awards ceremony gives people throughout the U.S. the opportunity check out some of Broadway's productions.
Dennehy captured the Tony for his powerful performance as the tragic Willy Loman, beating out Kevin Spacey of The Iceman Cometh.
Did Dennehy expect to win?
"To be honest with you, I thought Spacey would win," he replied.
Salesman director Robert Falls topped Howard Davies of The Iceman Cometh and Trevor Nunn for Not About Nightingales.
Elizabeth Franz accepted the Tony for best featured actress in a play for her portrayal of Loman's wife Linda. "I am truly possessed by this woman," she said.
Runaway favorite Dench won the prize for best actress in a play for her role as a theater diva in David Hare's Amy's View. Just months ago, Dench won an Oscar for best supporting actress for Shakespeare in Love.
"It is coincidental that I did Shakespeare in Love when I did, that I said I would come back [to Broadway] when I did," the actress told Mondale. "It's been coincidence that those things happened at the same time."
There is almost nothing more American than a good Broadway musical, and Short and Broadway favorite Bernadette Peters took top acting honors. Critics complained about the lac of successful new musicals, and Sunday night's trophy for best new musical went to a show that is a review of the old work of Broadway legend Bob Fosse.
Parade, a huge commercial failure about the lynching of a Jewish factory executive in the South in 1915, won Tonys for best original score and best book of a musical, which honors the unsung script in a musical.
The musical closed in New York after a brief run, but book writer Alfred Uhry, in accepting the award, said it would reopen for a national tour in June 2000 in Atlanta.
Heavily favored Kristin Chenoweth won best featured actress in a musical for her lively role as Sally in You're a Good Man, Charlie Brown, and Roger Bart won best featured actor for his role as Snoopy.
Frank Wood of Side Man won best featured actor in a play.
Swan Lake brought Matthew Bourne two Tonys for best direction of a musical and for best choreography. His director award was viewed as an upset, as Broadway veteran Hal Prince had been favored for his direction of Parade.
In accepting his award, Miller adopted a serious tone to remark that Salesman or Iceman, if written today, would be deemed too expensive or risky to be produced on Broadway.
Asked about the show's competition with another masterpiece, Iceman by Eugene O'Neill, Falls said backstage, "It's apples and oranges ... Somebody had to make a decision."
The 53rd annual Tony Awards returned to a Broadway theater, the Gershwin, this year, because Radio City Music Hall, its home for the past two years, is being renovated.
The ceremony also featured a series of presenters instead of one host. Rosie O'Donnell, who did those chores for two years, declined this year because of her busy schedule.
The Tony winners in 21 categories were chosen by more than 800 voters, including members of the theater community and theater journalists. The Tonys are named after Antoinette Perry, founder of the American Theater Wing, a theater service organization.