He was taken to Venezia by a wealthy patron when he was only 14 - Pier Francesco was an outstanding singer - and took on the patron's name, Cavalli, as a sign of gratitude.
He started as organist at San Zipolo, then was contracted as a singer and later on as an organist at San Marco. Finally he became maestro di cappella and thus Monteverdi's successor. He died in Venezia in 1676 as a wealthy man.
Cavalli has, of course, met Monteverdi. After all, Monteverdi was his boss at San Marco, and they worked together on a few other occasions, too. It is obvious that Cavalli was influenced by him who was already seen as the greatest composer of the time.
After Monteverdi's death, Cavalli succeeded him on that throne. He was even invited to Paris to write a festive opera for the wedding of Louis XIV, Ercole Amante - a rare honour indeed, but for some obscure reason, he was almost completely forgotten until not so long ago.
Cavalli's operas carry on the master's style although his later works reflect a general development towards what would become the typical style of baroque opera, i.e. a regular succession of recitativo - aria - rectitativo - aria etc.
This was not exactly Cavalli's cup of tea - like Monteverdi he had never clearly separated aria from recitativo and arioso which would have crippled the realistic rendering of emotions -, so he failed to completely conform to the new style and, towards the end of his successful life, - guess what? - went out of style.