The German lapped the scenic Red Bull Ring in a track record one minute 04.070 seconds, a mere 0.029 quicker than fellow-four times world champion Hamilton in the Mercedes, with sunshine replacing overcast skies.
Verstappen, meanwhile, parked up by the side of the track some 10 minutes from the end of the third and final session before qualifying.
"The engine just shut down," the Dutch 20-year-old, who had an army of orange-shirted fans packing the grandstands and is eager to make up for crashing out on the first lap last year, said over the team radio.
"Hopefully it's as simple as a box being changed," Red Bull principal Christian Horner told Sky Sports television.
Hamilton was quickest in both of Friday's practice sessions, at a track that has crowned a Mercedes winner in each of the four years since Austria returned to the calendar in 2014, but Ferrari appeared to have found some extra speed overnight.
Finland's Valtteri Bottas, last year's winner from pole position after team mate Hamilton took a grid penalty, was third on the timesheets with a time 0.134 seconds slower than the Ferrari.
The top three teams again filled the top six places, with Ferrari's Kimi Raikkonen fourth fastest ahead of Verstappen and Red Bull's Australian Daniel Ricciardo in sixth place.
France's Romain Grosjean, still without a point this season, was seventh with Haas team mate Kevin Magnussen eighth and Renault's Carlos Sainz ninth.
Sauber's Monegasque rookie Charles Leclerc continued his fine form in 10th, with Swedish team mate Marcus Ericsson providing a benchmark in 16th.
Leclerc ended the session as a spectator, however, after also having problems.
"I think there is something wrong with the engine," he reported.
(Reporting by Alan Baldwin, editing by Ken Ferris)
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