Spanish factory Ducati rider Marquez won the 11th sprint out of 12 this season to raise his tally in the championship standings to 356 points.
The 32-year-old leads his younger brother Alex by 95 points and third-placed Italian teammate Francesco Bagnaia by 156.
Factory KTM rider Acosta, 21, came home second ahead of Italian Enea Bastianini on a KTM-Tech3.
Starting from second spot on the grid, Marquez quickly eased past polesitter Bagnaia, still in the opening lap.
Acosta shot ahead from the third row at the start, passing Bagnaia halfway through the race to get just behind Marquez.
He glided into the lead moments later as Marquez slowed down in uncertainty over tyre pressure.
“I saw that the pressure was not enough. I tried to push some laps on the brakes, but I saw that was too much risk. I decided to wait,” said Marquez.
It took the six-time MotoGP champion a while to warm the tyres up and take the lead back with a swift overtake early into the last but one lap.
“When I saw that the pressure was inside the rules, then I pushed the last laps,” Marquez said.
Battling a forearm injury earlier in the season, Acosta relished his podium finish.
“It was the darkest beginning of the season of my life, and being on the podium again is super nice,” Acosta said.
Bastianini was equally happy with third, a week after sitting out the German GP over food poisoning.
“It’s incredible to be here. Now we are here to enjoy that podium, my first one with the KTM,” he said.
Reigning world champion Jorge Martin returned to the sport with an 11th spot.
Martin sat out the first three races following two pre-season crashes, and when he returned at Qatar in April, he crashed heavily again and missed the next seven races.
Marquez won the last four grand prix, taking the honours in both the sprint and the main race each time, and he will be looking to add another perfect weekend at Brno.
He is eyeing his seventh MotoGP world title – and first since 2019 – that would put him level with the legendary Valentino Rossi. (AFP)