Ducati star Bagnaia coasted across the line ahead of Aprilia's Maverick Vinales in the Algarve sunshine to follow up his win in MotoGP's inaugural sprint 24 hours earlier.
Marquez crashed and wiped out home favourite Miguel Oliveira on lap three to leave the way clear for Bagnaia to get his championship defence off to a perfect start.
Bagnaia, whose teammate Enea Bastianini missed the race after injury in the sprint, leaves Portimao with a maximum 37 points from the first of the 21-race season.
"That's the start me and my team wanted," the Italian told motogp.com.
"We have to keep going like this."
He then joked that after Saturday's 12-lap dash in the sprint "I'm not used to such a long race".
Bagnaia's compatriot Marco Bezzecchi, riding Ducati's VR46 bike, came in third to complete the podium.
If it was reasonably plain sailing for the world champion it was anything but for Marquez.
A rare moment of madness from the Spaniard provided the main drama of the afternoon.
The pole sitter clipped Jorge Martin and then his out-of-control Honda slammed into the back of Oliveira's KTM, wiping out the Portuguese rider who had high hopes of a podium finish in front of his home fans.
Marquez received a hostile reception from some of the fans on his return to the pits where he quickly made his way to Oliveira's garage to offer his apologies.
The six-time former world champion suffered a suspected hand fracture as well as a time-penalty, the last thing the 30-year-old needed after his last three seasons plagued by injury.
Martin, facing an uphill task to get back into the race, eventually slipped out with six laps left.
Oliveira had led fleetingly, following a quick start from the second row on the grid, before Bagnaia forged to the front, seconds before Marquez made his intervention.
Vinales was encouraged by his promising start to the season.
"I feel good, I missed the chance to overtake because 'Pecco' (Bagnaia), I knew, had a little more (pace).
"I'm actually really happy - we are going to fight in the front."
Daniel Holgado, 17, gained his maiden win in the Moto3 race earlier, with Pedro Acosta holding off Aron Canet for the Moto2 spoils.
Holgado's win came after a bitter sweet weekend for the KTM Tech3 team after Pol Espargaro's horror crash in practice on Friday left the Spanish veteran with "severe spinal trauma and a contusion to his lungs".
MotoGP heads to Argentina next weekend for the second race of a season that features two new venues on the calendar in India and Kazakhstan and the new sprint format on every race weekend.
