Haydn more or less invented sonata form, which is the structural principle behind not only piano sonatas, but also string quartets and symphonies.
What did Monteverdi do that in any way puts him into this same category? Well, he did pretty much invent opera. There were a couple of people that preceded him, but their efforts have been largely forgotten. The first real opera in European history is Monteverdi's L'Orfeo written in 1607. It is fairly clear that Monteverdi was one of the key creators of the Baroque in music. There is a very good argument that can be made for the honor being shared equally with Arcangelo Corelli who laid much of the groundwork for the Baroque uses of functional harmony as well as sonata and concerto form. Monteverdi's work falls into the three categories of madrigals, church music and opera. So it is fair to say that Monteverdi laid the foundations for vocal music in the Baroque, while Corelli did the same for instrumental music.
But Monteverdi played a unique role in that he not only was a master of late Renaissance vocal music, especially in his earlier books of madrigals, but he enacted the transition to the Baroque in his own music which was, in the later books, truly Baroque in style. This is much rarer than you would think in music because it means that Monteverdi was a master of not one, but two quite different musical styles. The closest parallel I can think of is C. P. E. Bach who was a transition from the music of his father, J. S. Bach, and the early classical style of Haydn. But C. P. E. Bach was neither a great master of the Baroque, nor of the Classical styles but a somewhat eccentric, though interesting, sub-category in himself. Monteverdi however was very much in the mainstream in both the late Renaissance and the early Baroque.
Just to sample a bit the two styles, let's listen to the first madrigal from Bk I, "Ch'io ami la mia vita":
There is nothing there that really hints at Baroque style: it is clearly within the bounds of the late Renaissance. You will hear a few ornaments that the performers add, but this practice dates far back into the early days of the Renaissance (and probably earlier, but we don't have too much evidence). Here is the first page of the score:
And there it is: the basso continuo and a prominent treble voice. Of course, later on the other voices join in as well, but the basso continuo is throughout, indicating the huge increase in the importance of and compositional command of pure harmony as an expressive device. Let's have a listen to this piece:
It begins with a short sinfonia for the strings and continuo alone. I just quoted from where the voice enters.
I think you will agree that this madrigal occupies an entirely different musical realm from the first one!