Kensington Palace thanked the public for "all of the support" and said that they were "delighted to be able to share this happy news".Both the Duke and Duchess are on a 16 day tour of Australia and New Zealand.
British media said Meghan, 37, was believed to be about 12 weeks pregnant.
Australian Prime Minister Scott Morrison was also present and congratulated the Duke and Duchess on their baby news.
The news comes while Markle and Harry are on their first major global tour since marrying in May.
Meghan said the koalas were "so, so sweet", while Harry reached out to pet one.
They are now on a tour of Australia, Fiji, Tonga and New Zealand.
Kensington Palace has confirmed no changes to the planned itinerary have been made despite Fiji and Tonga being listed as countries that pose a risk of Zika transmission; Zika is a virus that can cause birth defects.
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He added that Saudi Arabia will not hesitate to form a joint investigation committee with Turkey to uncover the truth. He said the incident is being investigated and pledged "very powerful" consequences if Saudi Arabia is found guilty.
The royal couple waved as they drove through the gates of the governor-general's residence in Kirribilli, just after 8am.
NSW premier Gladys Berejiklian said she looked forward to welcoming the couple in Sydney, sending her well wishes via a tweet: "Very pleased to hear the news that their Royal Highnesses, The Duke and Duchess of Sussex are expecting their first child".
"I had it engraved on the back, 'To M.M. From M.M, '" she added. She reacted jokingly, saying "I'm sure at some point we'll need the whole (lot)".
Harry has become immensely popular in Britain, in part because of his military service and tireless work on behalf of wounded soldiers, and he has spoken often in recent years of his desire to settle down and start a family.
In remarks which went from honest to humorous, the duke spoke of his pleasure at being able to show off Australia to the duchess. Harry has been patron of the WellChild charity since 2007.
He told ITV's Good Morning Britain in an interview a month after the wedding: "She's wanted children for a long time and when she met Harry and she spoke about how much she loves him, there's got to be a child in the making, somewhere soon".