The Spaniard started fifth but found himself bullied back to seventh on the opening lap before racing his way back to a podium position.
That included catching and passing the Mercedes of Lewis Hamilton and Ferrari's Carlos Sainz, the latter of which he did on Lap 45 after an entertaining wheel-to-wheel dice for the better part of a lap.
“It is obviously a perfect start for this project,” said Alonso after only his second podium since 2014.
“We didn't expect to be that competitive.
“I think the aim in 2023 was [to] get in the mix in the midfield, maybe leading the midfield and get close to the top three teams eventually.
“Even a podium was not in the radar in 2023, and we found ourselves second best car today in Bahrain, or the whole weekend, just behind Red Bull.
“This is a little bit of a surprise.”
Aston Martin emerged as a dark horse in pre-season testing.
Though its single-lap pace didn't set the world alight, it showed well when it came to race runs.
However, with different programmes, fuel loads, engine modes, and even tyres in play last week, getting an accurate read was difficult.
Even still, Aston Martin raised eyebrows with its performance and has seemingly confirmed that potential this weekend.
There was a degree of luck involved, with Alonso tagged by team-mate Lance Stroll on the opening lap, but the car was also rapid.
“I thought it was George [Russell] but I saw the replay on TV later and it was Lance,” Alonso recounted.
“He had a very good start because he was alongside me into Turn 4.
“We got lucky,” he added.
“Obviously two cars didn't have any problem and we could continue.
“It was our lucky day for many things; for this contact, and also to be both cars with a strong points.
“I'm very happy for the team because they deserve it.”
Aston Martin sits second in the constructors' championship with 23 points to its name, seven clear of Mercedes in third but already 20 adrift of Red Bull.
Alonso is best of the rest in terms of the drivers' competition, with only the two Red Bull pilots ahead of him in the standings.
The Formula 1 season continues with the Saudi Arabian Grand Prix on March 17-19.