Acosta finished third behind polesitter Enea Bastianini after Saturday's sprint winner Maverick Vinales, who was in second place, crashed on the final lap due to what appeared to be a technical problem with the gearbox.
Martin, last year's title runner-up, took the championship lead from reigning champion Francesco Bagnaia, who crashed out of the race following a collision with six-times MotoGP champion Marc Marquez on the Gresini Ducati.
Martin has 60 points after two races while Brad Binder is second, 18 points behind. Bagnaia was pushed down to fourth in the standings as a result of the crash, 23 points behind Martin.
"I knew what I was capable of. I started fast and I was trying to manage the tyres. Maverick and Enea were super close but at the end I saw a gap and it was enough to fight for the win," Martin said.
Martin rocketed off the line to take the lead going into turn one and he immediately set the pace for the race out front, clocking the fastest lap and extending his lead over Vinales.
Bastianini sat third with Bagnaia and Marquez behind him as the top five riders peeled away from the rest of the pack.
ACOSTA'S CHARGE
But behind them, GasGas Tech3's Acosta was unwilling to sit back and conserve his tyres, first overtaking Red Bull KTM's Binder before squeezing past Marquez to move up into fifth with a fastest lap of his own.
Acosta gave Bagnaia chase and attempted risky overtakes on the Italian and his persistence eventually paid off with five laps to go when he lunged on the inside and shut the door on the two-times champion.
Tyre management proved to be his undoing in the season opener in Qatar but this time Acosta looked comfortable with tyre temperature as he pulled away from Bagnaia and Marquez.
Acosta's heroics forced Bagnaia and Marquez to fight for fifth but with three laps to go, desperation ended in frustration as both riders collided and crashed when the factory Ducati rider attempted to retake his position on the inside.
Acosta looked set for a well-deserved fourth place but he was gifted a podium finish when Vinales crashed on the final lap.
In just his second MotoGP race, Acosta became the third-youngest podium finisher in premier class history.
"The bike was perfect. Today there was no problem with the tyres," Acosta said Acosta.
"We need to improve but look at this podium - it's not red Ducatis, it's a red GasGas!"
