But the Silver Arrows responded in the second running, with Bottas finishing just 0.027 seconds clear of Hamilton, and Verstappen third, three tenths adrift.
Hamilton will be keen to carry over the momentum from his controversial win at Silverstone a fortnight ago to the concluding round before Formula One's August shutdown on Sunday.
The Mercedes driver is only eight points behind Verstappen and the Silver Arrows' form in practice will provide Hamilton with the hope that he can record a second win in as many races in his record-breaking quest for an eighth world championship.
The slow and twisty circuit, a dozen miles outside of Budapest, is one Hamilton has enjoyed in his career and a win would rank as his fourth in succession in Hungary.
He is also attempting to become the first driver to win at the same venue on nine occasions, and the first to reach three figures with a landmark century of triumphs.
Hamilton ran wide at Turn 11 but escaped without damage, while Verstappen complained about the handling of his machine with a three-tenth gap perhaps cause for concern for the Dutchman and his Red Bull team.
Alpine's Esteban Ocon finished fourth, one spot ahead of Verstappen's team-mate Sergio Perez, who was eight tenths off the pace.
Pierre Gasly took sixth for Alpha Tauri ahead of double world champion Fernando Alonso and four-time champion Sebastian Vettel.
British driver Lando Norris, on an impressive run of 15 consecutive points finishes – the current longest streak in the sport – finished ninth for McLaren.
Earlier, Yuki Tsunoda crashed out of first practice after losing control of his Alpha Tauri through Turn 5.
The Japanese rookie missed the majority of the second running as his team repaired his wounded machine.