As a teenager, Tartaglia danced as Cookie Monster and Oscar the Grouch at Sesame Place. At the age of 16, while visiting the set for the taping of the 25th anniversary special, he was given a small part right-handing for John Kennedy in one scene. He subsequently began on Sesame Street when he was 18, performing background characters and assisting. Tartaglia was a regular puppeteer for the series through Season 35, though he has returned occasionally for different projects.
Tartaglia has since channeled his theatre and television experience into collaborations with The Jim Henson Company. In 2012, he conceived, directed, and hosted the Carnegie Hall concert Jim Henson's Musical World. He has also directed live stage show versions of Dinosaur Train and Sid the Science Kid, the cruise shows Jim Henson's Inspired Silliness and The Secret Silk, the Sesame Place stage show Elmo the Musical Live!, and the puppet direction for the 2021 mounting of Emmet Otter's Jug-Band Christmas. Tartaglia's television credits with Henson include Splash and Bubbles (series creator and voice of Splash), Julie's Greenroom, and Word Party.
In 2013, Tartaglia took over the role of Gobo Fraggle, starting with promos for The Hub and continuing through subsequent live appearances. In 2020, he scripted and executive-produced the Fraggle Rock: Rock On! shorts. In addition to reprising Gobo, Tartaglia puppeteered Boober and Uncle Traveling Matt (to Dave Goelz's voice), and Wembley (voiced by Frankie Cordero). He serves a similar role as executive producer, writer, performer, and puppet captain for the full-length series Fraggle Rock: Back to the Rock.