The royals were greeted by Dean and Chapter of Westminster, before taking their seats inside.
Harry and Meghan's engagements will include official meetings with Taoiseach Leo Varadkar and President Michael D Higgins.
They will also visit some of the city's main cultural attractions, including Ireland's oldest university, Trinity College, and Croke Park, where British troops opened fire on a crowd almost a century ago, on a day known as Bloody Sunday during Ireland's War of Independence.
They were due to attend a summer garden party at the British Ambassador's residence in the south of the city later on Tuesday, where they will meet people from across Ireland's arts, sports, military and social enterprise sectors. It was 100 years ago on April 1 1918 that the Royal Flying Corps and the Royal Naval Air Service merged to create the RAF - the world's first independent air service.
The couple will see the Book of Kells at Trinity College Dublin and will visit the city's starvation memorial.
As part of their two-day visit, the duke and duchess will visit Croke Park, Ireland's largest sporting arena and the country's main Gaelic football stadium, as well as the Irish Emigration Museum, Trinity College Dublin and a small start-up called DogPatch Labs.