The first time Connie the Hormone Monstress takes the screen in “Big Mouth,” she takes it over completely. She emerges from the shadows in a flash of lightning, as if zapped to Earth by some vengeful, horny god. As she takes in her new charge, a tiny redhead scared to death of her changing body, she grins wide enough to show her pointed teeth. Connie is a feat of smart animation before she says a word — but it’s once she opens her mouth and Maya Rudolph’s voice comes out that Connie becomes truly unforgettable.

It’s wild that “the Hormone Monstress” is the character who’s finally won Rudolph her first Emmy, but if you’ve seen even a minute of this performance, it isn’t shocking. Rudolph’s Hormone Monstress was a game-changer for “Big Mouth.” On a narrative level, her total commitment to the part ensured that Connie, and the teen girls looking to her for guidance, felt just as equal in stature to the boys ostensibly anchoring the action. As Connie dispenses confident advice, Rudolph counteracts Nick Kroll’s gravel-voiced Hormone Monster with a silken, lascivious smirk. She can (and often does) go from sentimental to furious to devastated within a single sentence. And as with many of Rudolph’s comedic turns, this character savors every syllable she gets. In her hands, banal phrases like “bubble bath” become undulating punchlines all their own.

The “Big Mouth” episode for which Rudolph won — “How to Have an Orgasm,” written by Emily Altman — has her helping a skeptical girl understand her own body, with an assist from a talking vagina played by Kristen Wiig. Connie’s attempts to provide such intimate guidance are absolutely ridiculous, but as Rudolph expertly finds opportunities in the dialogue to soften her character’s otherwise hyperbolic approach, it’s also strangely sweet.

At this point, Rudolph is a TV veteran who knows how to make every minute she’s onscreen count. She made such an indelible impression on “Saturday Night Live” during her seven years that the show well knows that bringing her back guarantees huge laughs. Her turn as a melodramatic TV host on “Up All Night” was so consistently the highlight of the show that it tried rebooting the entire premise around her. Amazon’s “Forever” found its best moments as Rudolph acted out the dreams and frustrations of an unfulfilled ghost. By the time she turned up as an annoyed, omniscient judge on “The Good Place” (a part for which she’s also Emmy-nominated this year), Rudolph had long since established herself as a performer who could make an immediate impression.

Rudolph’s ability to find the funniest, weirdest way to read a line is a singular skill — and it’s what landed her the first Emmy of her career. “Outstanding Voiceover” might not be the category she might’ve expected to help her reach this milestone, but looking back at the list of winners, it’s also frankly refreshing to see someone outside the “Simpsons”/”Family Guy” universes get recognized here. (Not a single other show has won in this category since its 2014 inauguration.) Animation has seen some extraordinary innovation recently, and Rudolph’s win is an encouraging sign that the Academy might not just be willing to look outside the genre’s typical honorees, but also reward some fresher competition that richly deserves it.