"The car seems competitive, this is good news for us. But it's only Friday and we need to keep working," Alonso said.
Alonso was cleared to race on Monday when F1's governing body overturned Renault's one-race ban stemming from a safety incident at the Hungarian GP.
"We still need a couple of tenths to win a Grand Prix," Alonso said. "But we are just behind that first group and if they make any mistake, we can take the opportunity."
Badoer, replacing the injured defending champion Felipe Massa, finished 18th with a lap time 2.6 seconds off Alonso's. Despite the time, Badoer said he was "happy with the way things went."
Badoer forgot to slow down in the pit lane, however. He received three separate speeding fines totaling $7,700. He also had to meet with stewards to explain his actions after a fourth offense.
Romain Grosjean's debut for Renault was smoother as the 23-year-old Frenchman -- the first to drive for the French team in 24 years -- finished 13th, 1.383 seconds off his teammate's pace.
Meanwhile, Button has seen his championship lead slip. He leads Red Bull's Mark Webber by 18.5 points following three poor races.
McLaren's Lewis Hamilton, who was coming off his first win of the season at Budapest, was third fastest in the morning session. But his second stint ended after three laps when he damaged his front wing and lacked spares.
Badoer trailed leader Barrichello by 3.380 seconds after the first session as Michael Schumacher -- who was poised to replace Massa but was sidelined by an old neck injury -- watched from pit lane.
Qualifying on Saturday will determine the starting order for Sunday's 57-lap race.