While the way the Earth is saved by microorganisms seems naïve perhaps today, I suspect that the idea of microorganisms affecting aliens was very new since it was not even completely accepted that disease was caused in humans by bacteria (or had only recently been accepted) at this time.
Did any previous story feature microorganisms or was Wells the first, as he often was, to come up with this idea?
A few years previously (1892) was The Germ Growers: An Australian Story of Adventure and Mystery, by Robert Potter. A hostile alien race mutates naturally-occurring Earth bacteria in order to conduct bio-warfare against humanity:
But this Davelli has lately taken up a line of action against God and man which some of the more powerful of his kind took up ages ago with far wider success; he has established here, and in the inaccessible parts of the Himalayas, and in one or two other places, artificial seed-beds of pestilence. His emissaries gather, from all quarters, germs of natural and healthful growth, and submit them to a special cultivation under which they become obnoxious and hurtful to human nature. And then they sow them here and there in the most likely places, and thus produce disease, death, and disaster among men.
The story is public domain, and available at Project Gutenberg.
After googling this, I found it interesting that Santiago Ramón y Cajal wrote an anthology that could be regarded as 'science fiction' in 1885-86 named Vacation Stories. From the essay I came across:
Cajal’s first story, “For a Secret Defense, a Secret Revenge,” describes the strategy of an aging bacteriologist who suspects his young wife is having an affair with his lab assistant. After assuring himself of their guilt, based on the print-out of a seismographic device rigged to the laboratory couch, he infects the assistant with bovine tuberculosis bacillus, waits to see whether it will be transmitted to his wife and then writes up the results in a bacteriological journal.
I don't know if this fits the scope of your question, but it appears to pre-date even the above answer about Robert Potter.
Professor Bommsenn's Germ by Ernest George Harmer is an option, a short story published in 1887. The story is available on the website technoccult here. A short description from the site is
A scientist uses electricity to evolve a germ into an evolved human. The mutant human has no teeth or hair and possesses some sort of mesmeric ability. A “spoof of current biology.”
Found using the search feature on the isfdb.org site from DavidW's answer, with the search query 'title contains germ, title does not contain german, sort by date'.
A couple of years earlier than my previous answer (25 Janvier 1980) is "Professor Bakermann's Microbe" (original title "Le microbe du Professeur Bakermann, récit des temps futurs") by Charles Epheyre, first published in Revue Bleue (Revue politique et littéraire). (Also noted in the annual index.) Note that ISFDb records an incorrect date and source of first publication; it is correct in Stableford's anthology Scientific Romance: An International Anthology of Pioneering Science Fiction (2017), which is where I found an English translation. The full story is currently part of the Google books preview of Scientific Romance.
In the story the titular Professor Bakermann, living in the future world of 1935 where the microbes causing all known diseases have been isolated, dreams of creating a more powerful disease that can't be stopped. He breeds a "microbe" ("virus" is also used in the story) that is resistant to all known drugs and treatments, which naturally gets out of his lab when his wife violates its sanctity to search for incriminating letters. Hundreds of thousands of people die, including half his town, before he discovers it is susceptible to an electrical field. He becomes a hero and with ample rewards and, freed from his wife to do what he wishes, he sets out to breed an even more unstoppable disease.
In 1935 all diseases have been identified and classified:
Now, in 1935, the works of Pasteur had long been surpassed. Obedient to the impulse provided by the master, all the scientists of Europe, America, Australia, and even Africa, had set to work. Thanks to them, the most difficult problems had been clarified, the most obscure problems resolved; there was no longer any disease that did not have its microbe, labeled, classified and stored. The forms, the behavior, the habits and the tastes of all terrestrial, marine and airborne microbes were known, and microbial science had become the basis of medicine in all the universities.
But Bakermann has bred a new, much more dangerous microbe:
The means of rendering harmful microbes inoffensive were known, but that was only one aspect of the problem. Bakermann had found a means to render inoffensive microbes harmful.
When we say "harmful" we do not mean to imply mildly harmful, but terrible, overwhelming and irresistible. The microbes presently known only kill in a day, half a day at worst, and are also possessed of a fragile vitality. It does not take much to attenuate them or render them harmless. The problem, therefore, was to have a virus powerful enough to kill in an hour, at a dose of a hundredth or thousandth of a drop, in such a manner that no living creature could survive it. Above all - and this was the most delicate part - the terrible microbe must be very resistant, incapable of allowing itself to be weakened by intemperances of climate or the medications that artful humans were inventing incessantly.
[...]
We cannot enter into detail here regarding the famous scientist's scientific techniques. At any rate, thanks to improved culture media and certain electrical procedures that he was still keeping secret, Bakermann had profoundly transformed a vulgar microbe, the microbe that turns butter rancid - very widespread, alas! - by submitting it to a whole series of complicated cultures, and he had made it into an extremely nasty microbe.
The virus spreads extremely rapidly:
The town of Brunnwald was half-annihilated. Vienna and Munich already counted a few fatalities, and were probably infected at all points. Paris, London, Rome and St. Petersburg were invaded, without anyone being able to prevent the invasion, and the current evaluation was that the entire human race would be doomed within forty-eight hours. It was enough to make the greatest heroes shiver.
And with a huge fatality rate, even as Bakermann perfects the electrical cure:
The telegraph brought frightful news with every passing minute. At the very moment when, thanks to the positive electricity platform, the population of Brunnwald was entirely reassured, there had been 45,329 deaths in Berlin, 7,542 in Vienna, 4,673 in Munich, 54,376 deaths already in Paris, and 58,352 in London.
In brief, there had already been a total of 684,539 deaths in Europe.
And finally a victorious Bakermann dreams of an even more dangerous bacillus:
Nevertheless, he finds some consolation in trying to make a better mortifulgurans, more vigorous, more invincible than the first, whose irresistible effects no electricity, nor any medication, known or unknown, will be able to combat.
