Eugenijus Tulauskas, from Lithuania, was up to four times over the maritime limit when arrested in September last year.
A pilot had to take control of the container ship to ensure its safe passage into harbour.
The 44-year-old seaman, of no fixed abode was fined Â£1,500 at Belfast Magistrates' Court.
Tulauskas had contested a charge of having excess alcohol while on duty as professional master of a ship.
His lawyers argued that he was not on duty at the time of the offence.
The court was told that an experienced pilot sent out to guide the ship into port thought he smelled alcohol on Tulauskas' breath.
He contacted Belfast Harbour Police who detained the defendant and took a breath sample.
Talauskas confirmed at that stage he had not taken any alcohol in the previous four hours.
The district judge said she was "satisfied on the evidence that the defendant was not just master of the vessel, but was on duty at the time".
Talauskas' lawyer argued that he was aware of the pilot's actions and had congratulated him on his manoeuvres.
"It's not a case that he was totally incoherent," the lawyer said.
The court heard that the captain had since lost his job.
The judge, who could have fined him up to Â£5,000, commented that "if the defendant were in employment I would be looking at the upper region (of a fine)".
A sea captain has been convicted of being drunk in charge of a merchant ship in Belfast Lough.