George Russell led a Mercedes 1-2 ahead of teammate Kimi Antonelli in the season-opener, crossing more than 15 seconds ahead of Ferrari's Charles Leclerc in third and Hamilton in fourth.
"I’m genuinely really proud of the team,” the seven-time world champion told Sky.
"I think we’ve done an amazing job to get the car to where it is. Of course we’re not as fast as Mercedes, we’ve got work to do, but we’re right in the fight."
Hamilton said Ferrari’s race pace was stronger than qualifying suggested, having started seventh on the grid.
"I think all weekend I’ve been really, really strong, but qualifying didn’t show the true pace," he said.
"We had a few problems through qualifying which meant that I was further back than I should have been."
Once the race began, Hamilton said he felt competitive from the outset and believed he might have had his first podium with Ferrari if the cars were on the track for a bit longer.
"It was a really, really fun race and it felt good for me,” he said.
"I was obviously closing the gap right at the end to Charles. A couple more laps and I think I would have had him — maybe one or two more laps."
While conceding Mercedes hold a clear advantage in the early season, Hamilton said the deficit was not insurmountable.
"We’ve got a lot of work to do to catch Mercedes, but it’s not impossible," he said.
"I do believe we can close the gap. It’s not going to be easy because it’s quite significant, particularly on a single lap. We need to find out whether it’s power or battery power.
"But the car is just as quick through the corners, so we’ve just got to keep pushing."