The Queen, who is known for her signature bright block colours and matching hats, wore a spring green coat by Stewart Parvin and matching hat by Rachel Trevor Morgan.
The duchess, a former actress on the TV show "Suits", is easing into royal duties after marrying Prince Harry last month at Windsor Castle.
In the end, it was Her Majesty who made way for Meghan quickly saying: "You go first".
The Royals then took a short vehicle journey to the Catalyst science museum in Widnes, where the Queen, wearing spring green coat by Stewart Parvin and matching hat by Rachel Trevor Morgan, unveiled a plaque to mark the opening of the £1.86 billion bridge over the River Mersey.
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Another substitute, Artem Dzyuba, made it 3-0 with about 20 minutes remaining, putting the game out of Saudi Arabia's reach. While the mood is dark in the Saudi Arabia camp, it could not be brighter for the hosts.
She added: "The Queen was the happiest I've seen her in a long time".
Abigail Grimes, a pupil at Runcorn's Beechwood Primary School, was chosen to present the duchess with a posy of flowers as it was the youngster's fifth birthday.
Meanwhile, Hello! pointed out that Princess Diana wore a very similar pair of earrings on several royal visits, suggesting the new Duchess has very similar taste to her husband's late mother.
For his next release, the Duchess chose a light Sussex slinky MIDI dress with a black small belt at the waist from the Givenchy.
For a woman normally full of confidence who walked down the aisle alone, Meghan appeared to be beset with nerves on this first trip out with her grandmother-in-law.
All eyes will be on the duchess, the first time she has joined the Queen for an event without her husband, as she gets to grips with her new role of ribbon cutting and plaque unveiling, under the watchful eye of the 92-year-old monarch.