"We're sorry to the team and drivers about the engine problems," said Honda Racing Development engineering director Shuhei Nakamoto after Sunday's race. "We will be getting a countermeasure in place in time for the Bahrain Grand Prix."
Briton Button, third in Malaysia last year where he secured the first podium of his Formula One career, retired after two laps on Sunday with a smoking engine.
Stand-in team mate Anthony Davidson, replacing Japan's Takuma Sato, quit on the same lap with an engine fire.
The team, part-owned by Honda, believed the problem was caused by an oil leak with a hurried countermeasure put in place proving insufficiently robust.
"We must now look forward, fully investigate the problem and I'm confident that, with the strength, skills and resources of BAR-Honda, we can soon turn the situation around," said team boss Nick Fry.
BAR, runners-up to Ferrari in the championship last season, have yet to score a point in two races.
Sunday's failure followed a strategic decision to retire both cars from the Australian season-opener to give the drivers new engines in Malaysia without incurring a usual 10-place penalty on the starting grid.
Under new rules, engines must last two successive races this year.
"I am incredibly disappointed and frustrated," said Button. "I felt I was one of the quickest cars on the circuit. In light of how our strategy would have played out versus the competition, we've let ourselves down.
"We have to sort this out in time for Bahrain."
Japanese rivals Toyota meanwhile celebrated their first podium since they entered the sport in 2002 and it was hard for technical director Mike Gascoyne, who had been strongly critical of BAR's exploitation of a loophole that has now been closed, not to feel smug.
"They are obviously going to be changing engines quite a lot, aren't they," he commented.
Toyota motorsport head John Howett said it just showed how difficult Formula One can be.
"Last year they had a fantastic season, they could do no wrong almost," he said. "This season I'm sure they've worked as hard as we have during the summer and they are really struggling.
"In the first place, I think it was wrong that there was a loophole that they could exploit because the spirit of the regulation was to run two races," he added.
"I think all of the rest of us understood we could do it but if we want to win, or score regularly, we've got to make the engines run two races. We don't think they did a favour to Formula One and the sport in what they did."
