The Briton finished runner-up to Red Bull's Max Verstappen (25) at Barcelona's Circuit de Catalunya with teammate George Russell (25) third in the first real test of their re-designed Formula One car.
"Mega job guys, Mega job," Hamilton shouted over the team radio at the chequered flag.
"The car felt great," he told Sky Sports television. "We've definitely got steps to continue taking to try and close the gap to the Red Bulls but this is way more than I expected this weekend and I thoroughly enjoyed the race.
"As long as I've been racing, if you're quick here you should be good elsewhere."
The points sent Mercedes second in the constructors' championship, ahead of Aston Martin, and the happy faces were a contrast to the gloom after Friday practice when Hamilton spoke of struggling to get in the top 10.
The team sent reserve driver Mick Schumacher (24) back to the factory for intensive simulator work on Friday night while Hamilton said he had stayed up until around 11 pm seeking solutions.
"We definitely didn't expect to have the result we had today," said Hamilton, who has yet to sign a new contract for next season.
Speaking to reporters, Hamilton said Mercedes would keep chasing down the Red Bulls - he finished 24 seconds behind Verstappen - and hoped to be challenging by the end of the year.
Red Bull have won all seven races so far this year, Verstappen five of them.
"I'm more focused on making sure we have the car next year to challenge them from day one," added Hamilton. "The further we continue to push the car this year, the more that impacts next year also in some ways."
Asked if he would be at Mercedes in 2024, he replied: "I haven't signed anything yet but I think we're meeting with (team boss) Toto (Wolff) tomorrow so hopefully we can get something done."
Wolff said it would probably take half an hour over coffee.
While happy with the team's change of direction after an early season wake-up call, the Austrian cautioned it was important to keep expectations real.
"You see where the benchmark is. It's at the end more than 20 seconds. Maybe it was 15 in reality but that's just not where we aim to be," he said.
"We just need to chip away...we are really good at grinding. Once there is a setup direction and a development direction, we just go for it."
