The tourists reached their target of 192 with three wickets remaining at Kensington Oval on a dramatic final day which saw Clarke boldly declare on 406-9.
Australia then bowled out the West Indies for 148 and Clarke was delighted with the result.
"This is as good as I've had, no doubt, because we had our backs to the wall for the first three days of the game," Clarke said.
"The spirit and the character of the guys in the change room is what drives you, I guess, as a captain to make a bold decision, to declare when I declared.
"The confidence around me from everybody in that group, there wasn't one bit of fear of losing that Test match - it wasn't spoken about. From day one of the Test, all that's been spoken about is what we have to do to win this Test match."
The second Test is due to begin in Port of Spain on Sunday.