He met Frank Oz at a puppetry festival in 1972, and during a vacation in New York City, he attended daily Sesame Street tapings. A few months later, Goelz showed his design portfolio to Jim Henson, and in 1973, he was offered a job with Henson Associates as a part-time puppet builder. His first assignment was to build puppets and design effects for a proposed Broadway show. However, the show was soon abandoned in favor of an ABC pilot, The Muppets Valentine Show, for which Goelz built characters and got his first crack at performing, playing Brewster, whom he also designed.

Upon Goelz's return to California, he learned that he had been replaced by his electronics employer, so he set up shop creating puppets and videos for industrial videos. Eight months later, in the fall of 1974, Henson offered him a full-time position as a builder/designer, and occasional performer in specials, while still allowing him to keep his industrial clients. Returning to New York, Goelz began work on The Muppet Show: Sex and Violence, for which he built the new host character, Nigel. Working from sketches by Jim Henson, Michael K. Frith, and Bonnie Erickson, he also built Animal, Floyd Pepper, and Zoot, the latter becoming his first major character.

In 1976, Goelz joined the rest of the Henson team and flew to London to begin work on The Muppet Show. In addition to reprising his role of Zoot and playing background roles, as in the earlier specials, Goelz was promoted to "Principal Muppet Performer" with the starring role of The Great Gonzo. The puppet had debuted in The Great Santa Claus Switch, as Cigar Box Frackle, and had made brief appearances in Muppet Meeting Films and Herb Alpert & the Tijuana Brass, with different performers. The sad-eyed creation was now given a permanent name and Puppeteer. However, in addition to playing Gonzo, Goelz was still employed in the Muppet Workshop.

As the 1980s progressed, in addition to switching between the manic Gonzo and the phlegmatic Boober (a variety which Goelz recalled as "stimulating"), Goelz played occasional new roles in specials, notably Rugby Tiger in The Christmas Toy:

Following Jim Henson's sudden death in 1990, and with Frank Oz continuing to focus heavily on directing, Gonzo the character and Goelz the performer gained increased significance, starting with the first new feature, The Muppet Christmas Carol. By performing Gonzo as Charles Dickens as narrator, Goelz (accompanied by Steve Whitmire as Rizzo the Rat, a pairing which would be repeated in subsequent productions) largely dominated the Muppet side of the film, and received top billing as "Muppet Performer" (a distinction which would continue through Muppet Treasure Island and Muppets from Space) "...when we did The Muppet Christmas Carol, he [Gonzo] developed a soulful side. He played the part of Dickens, and I just loved doing that. It just paralleled my own growth. Jerry Juhl wrote it as a way of getting Dickensian prose into the movie. But the fact that he chose Gonzo was very satisfying to me. And I think it was because he saw me changing and I think he felt that Gonzo could change too" (Film Threat). Goelz also took over the part of Waldorf from Henson. Muppets From Space was both Gonzo the character and Goelz the performer's first leading role in a Muppet production.

Apart from a brief stint operating the face of Earl Sinclair and performing hand-puppet guest characters on Dinosaurs, and reprising Rugby in The Secret Life of Toys, Goelz' most notable new television character was Stinky the Skunk in The Animal Show. Otherwise, the Puppeteer remained mostly occupied with Gonzo in movies, videos, and the 1996 series Muppets Tonight, the latter introducing a few new characters such as Randy Pig and Bill the Bubble Guy. Goelz also performed a handful of minor Sesame Street characters and appeared in The Adventures of Elmo in Grouchland as Humongous Chicken. His most recent credits include The Muppets' Wizard of Oz and the first few installments of the online series Statler and Waldorf: From the Balcony. In 2011 and 2014, he reprised his performance as Gonzo, Dr. Bunsen Honeydew, Zoot, Waldorf, Beauregard and other signature roles in The Muppets and Muppets Most Wanted. In 2015, Goelz voiced Subconscious Guard Frank in the Disney·Pixar film Inside Out.

Goelz recalled the hectic schedule of working full-time behind the scenes and in front of the cameras in a 2004 Film Threat interview: