Acosta fastest in final Emilia-Romagna MotoGP practice
Redbull GasGas rider Acosta clocked one minute and 31.451 seconds at Misano Adriatico, just 0.004sec ahead of former MotoGP champion Quartararo.
Reigning champion Francesco Bagnaia was 0.17sec off the pace in third as he prepares for his bid to overtake current leader Jorge Martin, whom he trails by seven points, in the riders' standings.
Pramac rider Martin was fifth behind Marc Marquez, who is also in with a shout of a seventh MotoGP title as he is 53 points behind his fellow Spaniard.
Marquez, who will join Bagnaia at Ducati's factory team next season, slid off the track towards the end of session but should be on the "Marco Simoncelli" circuit for the decisive second round of qualification at 11:15 local time (09:15 GMT).
The first points of the weekend will be up for grabs in the sprint race which starts at 15:00 (13:00 GMT).
