Poehler, the Golden Globe winner of NBC TV series Parks and Recreation, is the voice of Joy, Riley's first emotion, who arrives as soon as she is born.
But the central message of the film, according to the actress, is that it's OK to be sad.
"Joy has been in charge for 11 years and is pretty comfortable being in the driving seat and assumes she's going to be there forever," she says, "and we find out very quickly that Sadness, Anger, Fear, Disgust - the other emotions - want a turn at the wheel."
Poehler adds: "I love the message of the film which is basically that no-one can be happy all the time - and that's OK. In fact, sadness may get you where you need to be.
"It's a very revolutionary idea, especially for parents who are constantly obsessed with their children's happiness, and in doing so don't often allow them to be in the moment and feel their feelings, and it's a great thing for children to understand, too."
When Riley's family move from Minnesota to San Francisco, the trauma pitches Joy and Sadness out of headquarters, leaving Anger, Disgust and Fear in charge.
Or, in other words, according to director Pete Docter, Riley starts becoming a teenager.
Docter, a father of two, wrote and directed Pixar's double Oscar-winning film Up, as well as Monsters Inc, and named the young heroine of Up after his daughter Ellie, who also voiced the part.
"My daughter was very like the Ellie of Up - spirited and sparky," he explains. "But then as time went on, she became quieter and more withdrawn. It was really hard to know what she was thinking as she started to grow up."
Producer Jonas Rivera adds: "Inside Out is our attempt to answer Pete's question of 'What's going on inside my daughter's head?'
"We didn't know either. I think a lot of parents ask that question."
Each emotion is given a distinct colour and shape, from the green of Disgust - a homage to Riley's hatred of broccoli - to the red block of Anger, who emits fire when disturbed.
This, according to Poehler, who runs an online community for girls and women called Smart Girls, could be a step forward for children trying to articulate their emotions.
She says that, as the mother of two boys, she knows how hard it is for children to say what they feel.  "What [the film] does is give you the tools to talk about emotions and feelings which isn't very easy to do all of the time."
It would be satisfying, she says, to hear a child say, "I feel like Anger today" because he or she can visualise it.
However, Docter denies that Inside Out is Pixar "doing" psychotherapy for children - or for adults, who, he believes, could benefit just as much from the film.
"So many adults employ therapists because they can't express themselves," he points out. "As a kid, at least if you are angry you lie on the floor and kick and scream, but as we grow up we realise it's not socially acceptable to do that and sometimes we bottle things up.
"The theme of the film is that our emotions bring us together - storing things up and claiming to be happy all the time doesn't actually connect you with people on a deeper level."
But does the film accurately reflect primary human emotions?
Psychotherapist Dr Dacher Keltner, from the University of California, who was a consultant on the film, says what it gets right is the idea that emotions oscillate.
"There will be a time when your mind is filled with fear - a second or two before shifting to anger. The movie portrays that struggle over the control panel that I feel to be true scientifically."
And so far, Pixar's foray into psychology is winning audience hearts as well as their minds - not to mention their wallets.
Since its US debut a month ago, Inside Out is nudging the half-billion dollar mark (Â£320m) at the box office, putting it on track to become the most successful Pixar movie.
The Independent called it "an instant classic" while the Radio Times noted that "Pixar... never fails to locate the sweet spot between stylish smartness and tear-jerking goofiness".
But Docter thinks that the real success of Inside Out can't be measured financially. "It's a first for children at a pivotal age, and it's about someone relatable, not a superhero or fairytale characters.
"A movie that can tell little girls that it's hard to grow up and it's OK to be sad about it is actually profound."
Inside Out is released in the UK on 24 July
It's the final frontier, according to actress Amy Poehler - but the action in Pixar's latest animation, Inside Out, doesn't take place in space, but in the caverns of the human brain, as the emotions of an 11-year-old girl, Riley, go on an adventure inside her head.