The Brit set a blistering pace in the opening session, clocking a 1:05.975s - the fastest ever lap of Spielberg - on the Soft tyre to end up three tenths clear of Verstappen's Red Bull in P2.
Lewis then went even quicker in the afternoon, following up his record-setting pace from the morning with a lap of 1:05.483s - 0.147s faster than Vettel's Ferrari.
"It's been a really good Friday with no major headaches to complain about so far," said the three-time champion. We had to swap out a spark plug during FP2, but the guys did a great job to turn the car around and we still managed to complete our programme.
"Most importantly, the car feels fantastically fast here. There's already a nice balance and it feels good out on track. This car is so quick in comparison to what we raced here last year. It's tricky, but a proper thrill to hook up a lap.
"The team is in good spirits and we're all up for another exciting fight with the Ferraris this weekend."