What's the first sci-fi story where a pet has an important role?

I suggest that among the earliest science fiction stories are those of Jules Verne.

There are two dogs, Diane and Satellite, in Jules Verne's From the Earth to the Moon (1865) and Around the Moon (1870).

A dog named Top is a major help to the castaways in Jules Vernes's The Mysterious Island (1874–75).

Top is definitely a pet of one of the characters in The Mysterious Island (1874–75), and if the dogs in From the Earth to the Moon (1865) and Around the Moon (1870) are pets and not scientific test subjects, From the Earth to the Moon (1865) might possibly be the earliest example of pets in science fiction.

The Man in the Moone (1620s)

In this book, the protagonist tames some birds and uses them to pass messages.

Eventually, he realizes he can use those birds to power a flying machine.

He ends up flying to the moon.

Copernicus is mentioned by name, indicating that the author was clearly inspired by the recent scientific discoveries in Astronomy.

So as long as "tamed birds" count as pets, pets in scifi is pretty old.

The Last Man (1826)

If it HAS to be a dog or cat, then Mary Shelly's lesser known novel The Last Man might work.

The novel involves a global pandemic that ends human civilization. At the end of the story, a man and his dog are living in Rome, and it's been a year since they've seen another living human. They depart together to travel to a different continent in search of survivors.

1940: "Quietus", a short story by Ross Rocklynne, first published in Astounding Science-Fiction, September 1940, available at the Internet Archive. After an apocalyptic asteroid strike has wiped out most of humanity, the Last Boy, accompanied by his pet crow, is stalking the Last Girl. Angry with his crow for frightening the girl, he throws stones at it. An visitor from outer space, who happens to be avian, misunderstands the scene and kills the boy to save the crow.

Mentioned in the question but erroneously dated 1950.

1938: "The Smile of the Sphinx", a novelette by William F. Temple, first published in Tales of Wonder #4, Autumn 1938. I didn't find a copy of that issue online, but the reprint in Worlds Beyond, December 1950 is available at the Internet Archive. Domestic cats have a super-intelligent group mind that secretly rules the earth. Cats and dogs originated on the moon; the lunar craters are the result of an ancient war between cats and dogs; a superweapon devised by the cats reduced the intelligence of the dogs.

  "It certainly is odd," I agreed. "Well, what is yoar theory?"
  "This: that all cats are not so innocent as they appear. That they are an ancient and alien race, with intellects far greater than ours. That they are parasites of the human race, and move amongst us as unsuspected spies, hearing, seeing everything we do, yet never betraying themselves in any way. Believe me, they're the world's best actors! They know their role by heart — they've practiced it for thousands of years, and never yet made a slip."
  "What could possibly be their motive?" I inquired.
  "I do not think there is any evil intent: they are above evil. It just suits their convenience that we should make pets of them and keep their physical bodies alive for them, for their bodies really are only husks — they live their true lives in their minds. But these husks are necessary, for mind must have a living body to keep it supplied with energy.
  "The idea of reincarnation actually works for them, too. When a cat's body dies, the mind that inhabited it transfers itself to some newly-born kitten."

1942: "The Star Mouse", a novelette by Fredric Brown, first published in Planet Stories, Spring 1942, available at the Internet Archive. A scientist puts his pet mouse in an experimental rocket to the moon, which is intercepted by space aliens who uplift the mouse to human intelligence.

An early example is Simak's Desertion from November 1944, in which a pet dog has a vital role, as discussed here

Therefore, Fowler decides that the next person to go out into the Jovian atmosphere will be Fowler himself. Though he won’t be going alone. Instead, he’ll take his faithful dog Towser with him, because Fowler would feel bad about leaving him behind.

Fowler also realises why none of the people he sent out ever came back. Because life is simply so much better as a Loper, the surface of Jupiter is beautiful and there are so many mysteries to explore.

Towser declares that he won’t go back, because they would only turn him into a dog again. Fowler pities the people in the dome who have no idea how wonderful life as a Loper really is. But he also realises that he couldn’t live in his old human body anymore, not even for a short while, because its limitations would simply be too much to bear, now he knows how much better life can be.