Bezzecchi clocked the best time of one minute, 43.947 seconds to pip Pramac Racing's Martin by 0.043 seconds for his first pole since the British GP in August. Bagnaia of the factory Ducati team was a further 0.21 seconds behind.
Bagnaia holds a 36-point lead over Martin in the overall standings, with Bezzecchi a further 29 points behind in third.
Luca Marini, Bezzecchi's teammate, was fourth ahead of factory Honda duo Joan Mir and Marc Marquez. The sprint race takes place later on Saturday.
"It was really tough," Italian rider Bezzecchi said. "We struggled a bit yesterday. This morning I felt better on the bike... I am very happy for this pole position.
"I want to dedicate this pole to a friend of mine that unfortunately left us a couple of days ago."
The Buddh International Circuit, which hosted Formula One races between 2011-13, recorded temperatures of 33 degrees Celsius (91.4°F) with 72% humidity at the start of the qualifying session.
The sprint race later on Saturday, as well as Sunday's main race, has been shortened due to the conditions at the circuit on the outskirts of New Delhi, organisers said.
Gresini Ducati's Alex Marquez has been ruled out of the remaining MotoGP weekend after suffering a double rib fracture in qualifying, his team said.
He secured a Q2 spot but crashed on his last Q1 lap and did not return for the second session, before being taken to hospital for further examination.
