Alazraqui was born in Yonkers, New York, to Argentine parents and moved to Concord, California, at an early age. He later relocated to Sacramento to attend Cal State Sacramento where he began competing in open mic contests. After winning in his fourth year of competing at the San Francisco International Comedy Competition, he took his prize money and moved to Los Angeles.
Alazraqui has since performed several voices for Nickelodeon cartoons, including Rocko on Rocko's Modern Life, as well as Denzel Q. Crocker and Juandissimo Magnifico on The Fairly OddParents, Winslow on CatDog, and additional voices on Angry Beavers and KaBlam!. Alazraqui also voiced several additional voices for the Cartoon Network show, Cow and Chicken during its run in the mid-late 1990s. He also voiced the titular protagonist of the first game in the Spyro the Dragon series, the Taco Bell chihuahua, Rikochet in the first two seasons of Mucha Lucha and Mr. Weed (the head of the "Happy Go Lucky" toy factory) on Family Guy. It is mentioned in the DVD commentary track that Alazraqui was reluctant to leave the Family Guy series. In particular, Seth McFarlane suggests that the death of Alazraqui's character, Mr. Weed, took the actor by surprise.
In 2005, he added the voices for two Cartoon Network series - The Life and Times of Juniper Lee where he played Monroe, a guardian dog with a Scottish accent and two characters in Camp Lazlo: Lazlo and Clam. In 2006 added Salty Mike from Squirrel Boy and Walden in Wow! Wow! Wubbzy! on Nick Jr. to his credits. He played Wisk in Glen Hill's 2000 film version of L. Frank Baum's The Life and Adventures of Santa Claus. He also provided the voice of Nestor in Happy Feet.
Jeff "Swampy" Marsh, a storyboard writer for Rocko's Modern Life, described Alazraqui's normal voice as bearing "no accent at all". Marsh describes Alazraqui's "Scottish accent" as "one of the best" and that he performs his other accents as "all very well". According to Marsh, Alazraqui uses various accents in his stand-up comedy routines.
Alazraqui also does work for the Disney Channel on the Playhouse Disney show Handy Manny. He provides the voice for Felipe, the bilingual Phillips-head screwdriver, and for Abuelito, Manny's grandfather. He was also the original "voice guy" on the Stephanie Miller Show. His replacement was voice actor Jim Ward, who had previously substituted for him. In January 2008, he returned to fill in for Jim Ward briefly while he recovered from surgery.
Alazraqui is known for his live-action work in the mockumentary series Reno 911!, in which he played Deputy James Garcia for five seasons (2003–2008). He played the same role in the series' spinoff film, Reno 911! Miami. As a nod to this role, he also played a "bumbling" Reno cop in the CSI episode "Let it Bleed". He has been a celebrity on the Tom Bergeron version of Hollywood Squares.
Alazraqui has written and starred in The Last White Dishwasher, a short film co-starring Tara Strong. In 2009 Alazraqui collaborated with Ted Nicolaou on a web series "The Club" which was released in November 2010. "The Club" features Jill-Michele Meleán, Debra Wilson, Johnny A. Sanchez, Lori Alan and Daran Norris among others.