In a statement, the Palace said "recollections may vary" but the matters will be addressed privately.
Meghan told Oprah Harry had been asked by an unnamed family member "how dark" their son Archie's skin might be.
The Palace said the Sussexes would "always be much loved family members".
The response from Buckingham Palace came after crisis meetings involving senior royals.
The Palace had been under growing pressure to respond to the interview in which Meghan - the first mixed-race member of the modern Royal Family - said that questions had been asked about their son's skin colour.
Prince Harry later clarified to Oprah that the comments were not made by either the Queen or the Duke of Edinburgh.
In other developments, Piers Morgan has left ITV's Good Morning Britain show following a row over comments he made about the Duchess of Sussex.
The statement from the Palace, which came a day and a half after the interview was first broadcast in the US, said: "The whole family is saddened to learn the full extent of how challenging the last few years have been for Harry and Meghan.
"The issues raised, particularly that of race, are concerning. Whilst some recollections may vary, they are taken very seriously and will be addressed by the family privately.
"Harry, Meghan and Archie will always be much loved family members."
It is understood the royals wanted to carefully consider their response and to give the British public an opportunity to watch the interview first when it was broadcast on Monday evening.
The royals are said to consider this a family matter and to believe they should be given the opportunity to discuss the issues privately.
Although likely, it was never absolutely certain that the Queen would issue a statement following the interview given by her grandson and his wife.
The risk was that any response would add to the "Royals in crisis" narrative which is playing out across front pages and TV screens across the globe.
But having had time to discuss the claims made by Harry and Meghan, four short sentences convey the message from the Palace that this is a private family matter and the specific allegations will not be publicly addressed.
The tone is conciliatory, with reference to the Sussexes as much loved family members. There is a gentle challenge in the line "whilst some recollections may vary", but also an assurance that concerns will be taken seriously.
And we can be sure that - at least as far as the UK-based royals are concerned - any bridge-building will be done in private.
During a visit in London earlier, Prince Charles did not respond when he was asked if he had seen the interview, in which the Sussexes addressed deeply personal topics of racism, mental health, the media and other members of the Royal Family.
The UK broadcast on Monday night was watched by an average of 11.1 million people.
In it, the duchess said there had been times when she "didn't want to be alive any more" because she found royal life so difficult. She said she asked for help from the Palace but received none.
Meghan said conversations about how dark Archie's skin might be when he was born took place "in tandem" with discussions about why he would not be given a royal title and police protection.
Under rules in place since 1917, the couple's children would not automatically become princes or princesses - unless the Queen was to intervene.
Asked by Oprah whether there were concerns that her child would be "too brown" and that would be a problem, Meghan said: "If that is the assumption you are making, that is a pretty safe one."
But the couple have refused to say which family member made the comments. "That conversation, I am never going to share," said Prince Harry. "At the time it was awkward, I was a bit shocked."
Harry also said it hurt that no-one in his family ever spoke out in support of Meghan after news headlines and articles with "colonial undertones".
Labour leader Sir Keir Starmer said Meghan's allegations about racism and a lack of mental health support should be taken "very seriously".
Downing Street said Prime Minister Boris Johnson had watched the interview on Monday night, but declined to comment further.