Like his former Alpine team-mate Esteban Ocon in Bahrain, Alonso picked up a five-second penalty for being out of position at the start, lining up to the left of his second grid slot.
And not unlike Ocon, Alonso was investigated again for not serving his time penalty correctly during his only pitstop.
After a lengthy delay, the FIA stewards concluded one of Alonso's mechanics was touching the car before the five-second penalty window was up.
Minutes after heading to the podium to pick up what would have been his 100th trophy in F1, Alonso was hit with an additional 10-second penalty, which dropped him to fourth behind Mercedes driver George Russell.
When asked about his first five-second penalty, which was the result of being left of his box by some margin, the Spaniard said he wasn't sure yet how he made the mistake.
"I need to review obviously; I made a mistake" he explained before his second penalty was confirmed.
"When they told me five seconds penalty, I said okay, I need to drive a little bit faster to make [up] those five seconds. But I need to review."
Alonso said he had to put in "qualifying laps" to try and bring home third.
In the final laps, he was being put under pressure by Russell, because the Briton's Mercedes team was also expecting Alonso to be hit with another penalty and their gap was hovering around five seconds.
"It is hard a grand prix but, those guys [in the team] they gave me power and I push all the way through, like qualifying laps.
"Red Bull's maybe a little bit out of reach, but the rest were behind so I'm happy for that."
The second penalty meant Alonso missed out on scoring his 100th podium, four days shy of the 20-year anniversary of his first podium in the 2003 Malaysian Grand Prix.
Regardless of the late penalty, Alonso and Aston Martin confirmed their strong start of the season by keeping both Mercedes and Ferrari at bay on the track.
Alonso said his start to 2023 was "probably unthinkable one month ago when we launched the car, but these guys they've made a fantastic car."
However, team-mate Lance Stroll retired from the race after losing power on lap 18, causing the first and only safety car of the 50-lap race.
Aston heads to the Australian Grand Prix third in the constructors' standings, six points behind Mercedes and nine points clear of Ferrari.
UPDATE: Alonso penalty overturned
Later on Sunday night Alonso’s podium finish was reinstated after his Aston Martin squad won a right to review.
