Last year's ceremony never took place, reports CBS News Correspondent Steve Futterman: It was supposed to be held on Sept. 11, and was cancelled.
"We honor those we lost and those that helped us get back on our feet," declared co-host Gloria Estefan. "We love you, New York."
Sanz, from Spain, was up for four awards. He won album of the year honors for "MTV Unplugged" and record and song of the year for "Y Solo Se Me Ocurre Amarte."
Sanz is on a roll, correspondent Hattie Kauffman reports for The Early Show. Not only did he take the top three honors this year, but last year he also was the big winner with four Grammys.
Last year's song and album "El Alma Al Aire" was record, album and song of the year, as well as best male pop vocal album.
"It embarrasses me principally because all of the artists who are here deserve it as much as I do," Sanz said backstage in Spanish.
Then he added in English, hesitantly, "Two years in a row is too much."
Colombian singer Carlos Vives and Mexico's Ramon Ayala y Sus Bravos del Norte won two awards apiece.
Vives won the best contemporary tropical album award for "Dejame Entrar," which also earned him a Grammy in February, and shared the best tropical song honor, a songwriters' award, for the album's title track. He entered the competition with a leading six nominations, including record, album and song of the year for "Dejame Entrar."
Ayala and his group were honored in both categories in which they were nominated, best norteno album for "El Numero Cien" and best regional Mexican song for "Del Otro Lado Del Porton."
The third annual Latin Grammys, airing on CBS, got underway with a hot rendition of "Celos" by Marc Anthony, followed by a rousing duet performance by Colombian rocker Juanes and pop singer Nelly Furtado and a sizzling performance by Mexican singer Thalia who sang "No Me Ensenaste" on a stage blazing with pyrotechnics.
Latin veteran Carlos Santana and rap/metal rockers P.O.D. also performed a moving rending of a new song titled "America" from Santana's forthcoming and highly anticipated new album.
There was a bit of controversy, reports Futterman: A number of Cuban performers were not able to attend because of tougher immigration laws. A group of demonstrators gathered outside the gala at Hollywood's Kodak Theatre to protest Washington's visa denial for Cuban Grammy nominees.
Twenty-two nominees, including jazz pianist Chucho Valdes, guitarist Rey Guerra and rapper X Alfonso, couldn't get entry visas. Valdes didn't get the chance to accept his award in person for best pop instrumental album for "Canciones Ineditas" — his first Grammy after decades in the business.
The State Department in Washington declined to comment. But a U.S. official based in Havana, speaking on customary condition of anonymity, said Wednesday that a new law that took effect Aug. 1 requires more rigorous procedures for visa applicants from countries listed on the U.S. State Department's terrorism watch list, which includes Cuba. The process took weeks longer than expected and there wasn't enough time for the nominees to get their visas.
Colombian superstar Shakira won best video for "Suerte" — the Spanish version of her huge English hit, "Wherever, Whenever." And Banda Cuisillos won best banda album, for "Puras Rancheras con Cuisillos."
The Chilean rock group La Ley, also up for four awards, won for best rock album for "MTV Unplugged." Cuban-American singer Jorge Moreno, on Madonna's Maverick Musica label, won the best new artist award.
The Grammy for bestfemale rock vocal goes home with Mexico's answer to Madonna, Alejandra Guzman.
"I won't dye my hair or try to be an American because I'm not," she told the crowd. "I love my culture. I love my language, and that's why my music is in Spanish."
Vicente Fernandez won best ranchera album for "Mas Con El Numero Uno." It was the first Grammy of any kind for the Mexican singer, who's known as "the king of ranchera music" and has recorded 55 albums. He'd been honored Tuesday as the Latin recording industry's person of the year.
Most winners weren't present for the early awards, but Alejandra Guzman was — and she ran on stage, screaming and flailing her arms, to accept the best female rock album award for "Soy."
"Wow! It's great! I'm a believer!" said the Mexican singer-actress, a first-time winner. "I believe in God and I believe in the people that did this for me."
It was a bilingual ceremony for the most part, as presenters and winners went back and forth between English and Spanish.
"Excuse me, but I gotta talk to my people in Spanish," Shakira said in English before switching languages for her acceptance speech.
Presenters ranged from Hector Elizondo and Daisy Fuentes to Nick Carter of the Backstreet Boys and Justin Timberlake of 'N Sync - both of whom are about to release solo albums.
While the annual Grammy Awards in February recognize only a few types of Latin music, the Latin Grammys cover a broad spectrum. Awards were given in 40 categories, including Tejano, norteno, Brazilian contemporary and merengue. To qualify for a Latin Grammy, an album must be at least 51 percent in Spanish or Portuguese.