With flood warnings in force locally, organisers kept spectators out for safety reasons until after the session had finished.
The governing International Automobile Federation (FIA) decided, however, that it was safe for the drivers to venture out and the session started on schedule despite the poor visibility and heavy spray.
With Friday's second practice entirely washed out, and qualifying looking uncertain with heavier rain forecast for later, the final session could prove crucial for Sunday's potentially title-deciding race.
While qualifying was still scheduled for 1400 local, it could also be held on Sunday if conditions are too bad on Saturday. Failing that, final practice positions will be used to determine the grid.
If so, Hamilton will be even more the favourite to take his third title on Sunday.
The Briton lapped with a best time of one minute 59.517 seconds, with Ferrari's Sebastian Vettel second fastest on 2:00.380. Nico Hulkenberg was third for Force India and Finland's Valtteri Bottas fourth for Williams.
Hamilton needs to scores nine points more than Vettel and two more than his own Mercedes team mate Nico Rosberg, who was only ninth fastest, to clinch the crown.
In a further boost for the Briton, Vettel and Finnish team mate Kimi Raikkonen both have 10-place starting-grid penalties for an engine change that exceeded their allocation for the season.
Several drivers aquaplaned in the tricky conditions, with Rosberg breaking the front wing on his Mercedes as he went off and nudged into the wall. Vettel also spun later on.
"A lot of water but not so bad," was the initial reaction of Red Bull's Australian Daniel Ricciardo.
The National Weather Service on Friday placed most of the state of 27 million people under a flood watch, warning of torrential downpours through the weekend. [ID:nL1N12N196]
The race is scheduled for 2 p.m. local time, when the weather is expected to be better.
There are three more races remaining after Austin, with Mexico -- hit by Hurricane Patricia on Friday -- the following weekend.
"My main worry is for my country," Mexican driver Sergio Perez said after Friday's second practice was cancelled.
"Mexico is being hit by the largest hurricane in its history and all my thoughts are with my people. I am praying for everyone to be safe."
(Editing by Ed Osmond)
