NB: As stated in the edited form of the question. Sauron didn't "die" in the traditional sense, however his body died and his spirit carried on. This is covered here.
Lets go in order from The Lord of the Rings books, through to the Appendices, onto The Silmarillion and finally into the Letters. (Emphasis mine)
To begin with the Council of Elrond
I beheld the last combat on the slopes of Orodruin, where Gil-galad died, and Elendil fell, and Narsil broke beneath him; but Sauron himself was overthrown, and Isildur cut the Ring from his hand with the hilt-shard of his father's sword, and took it for his own.
FotR: The Council of Elrond
Elrond doesn't tell us much beyond "Sauron was overthrown". This doesn't tell us if he died before or after, but tells us that before Isildur cut the Ring off, Sauron had at least been defeated (this becomes more important later on).
In Appendix B: Tale of Years, Tolkien says that Sauron passes away after the Ring is taken from him.
Sauron overthrown by Elendil and Gil-galad, who perish. Isildur takes the One Ring. Sauron passes away ...
FotR: Appendix B, "Tale of Years"
This seems to suggest that after being overthrown, Sauron remained alive, however after Isildur takes the Ring, he then perishes.
This is again supported in The Silmarillion where Sauron is described as being vanquished after the Ring was taken from him.
Sauron himself came forth; and he wrestled with Gil-galad and Elendil, and they both were slain, and the sword of Elendil broke under him as he fell. But Sauron also was thrown down, and with the hilt-shard of Narsil Isildur cut the Ruling Ring from the hand of Sauron and took it for his own. Then Sauron was for that time vanquished, and he forsook his body, and his spirit fled far away and hid in waste places
The Silmarillion - "Of the Rings of Power"
However, in a letter to Milton Waldon, Tolkien contradicts what he'd said above and suggest that Gilgalad and Elendil had indeed slain Sauron and Isildur cutting the Ring from his hand freed his spirit from his body.
“It ends with the overthrow of Sauron and destruction of the second visible incarnation of evil. But at a cost, and with one disastrous mistake. Gilgalad and Elendil are slain in the act of slaying Sauron. Isildur, Elendil's son, cuts die ring from Sauron's hand, and his power departs, and his spirit flees into the shadows. [sic]
Letter 131 - Letters of J.R.R. Tolkien
Saying in the act of slaying Sauron does not necessarily mean they had slain Sauron, but merely suggest that they were in the act of doing so. I don't see, however, why he'd change his opinion in his Letters before publishing the finished Lord of the Rings 3 years later.