Massa ignored the request - which was delivered over the team radio and heard worldwide via the television feed - and finished seventh, one place ahead of the Finn.
"Our fans expect us as a team to let our drivers race, and that's the overriding disappointment," said Williams.
"For our fans we want to apologise for that because we didn't handle it in the best way, and to let them down is not the way we want to go.
"We didn't handle the situation for either of our drivers particularly well, so of course we've apologised to our drivers."
The order scratched several old sores, coming early in the season at a circuit where coded instructions to Red Bull and Mercedes drivers last year triggered a lingering controversy and talk about the pecking order within those teams.
The phrase used by Williams was also almost identical to an infamous one that Massa heard when he was leading the 2010 German Grand Prix while Fernando Alonso's team mate at Ferrari.
"Felipe, Fernando is faster than you," has haunted Massa ever since and the Brazilian, who did as he was told that day, has ranked it among the worst moments of his career.
Williams, who took only five points in a miserable 2013 season but already have four times as many on the scoreboard after two races, said the team would make sure things were done differently in future.
She said the team had simply been caught out by a scenario that arose on the day.
"Now we have procedures in place so if it does happen again then we are prepared for that and, although you can't prepare for every scenario, hopefully we've got things covered.
"The instruction that was given in the way it was given, considering history, wasn't the most advised. But engineers - sat on a pitwall that is a highly pressurised environment - are people and people can sometimes make the wrong calls.
"We're only human, at the end of the day," she added. "But as long as we correct them and hold our hands up, that's the most important thing."
Massa, who spent much of his time at Ferrari battling in vain against the perception that he was the number two driver, was happy the air had been cleared.
He said his relationship with the former champions he joined at the end of last season, bringing with him some important Brazilian sponsors, had not been damaged and there were no lingering tensions.
"I believe that what's happened on the last race won't happen again," he told reporters.
"Team orders is part of our sport but it needs to be part of our sport when it's necessary. Everybody here is intelligent enough to understand when it's necessary and when it's not.
"When I do something that is my mistake, I will be the first one to say sorry. And the team was the first one to say sorry as well."
Bottas was also confident everyone was now clear about procedures.
"Things are okay," he said. "There are some things which are quite confidential with the team's rules and how you play it. We don't want to get into the finer details. There are different scenarios, and everything's sorted."
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