Meghan, 36, appeared at ease with her majesty as she passed another royal milestone, days after her first balcony appearance on Buckingham Palace on Saturday, during Trooping the Colour.
She and the Queen opened the bridge over the Mersey and unveiled a plaque to commemorate the event.
As they sat together during the ceremony, they appeared to be sharing a joke, as Meghan giggled heartily in response to her grandmother-in-law’s comments.
The opening took place at the Catalyst Museum, which offers a bird’s eye view of the new crossing from the glass top observation gallery.
This was Meghan’s first event without her husband, Prince Harry, by her side.
Her first solo visit with the Queen also comes much sooner after her marriage into the family than her sister-in-law.
Kate, the Duchess of Cambridge, first accompanied the Queen on a visit to Leicester in 2012, during her Diamond Jubilee tour.
It took place in March, around 11 months after Kate and William married in April 2011.
Body language expert Judi James said Meghan showed some nerves during the visit, but there was warmth between her and the monarch.
Ms James said: “The Queen was the happiest I’ve seen her in a long time. They looked like naughty teenagers giggling together at one point.
“Meghan was still being very careful. She sat with her legs crossed at the ankles and was clearly seeking approval.”
Ms James also noted the Duchess repeatedly touched her hair, a “self-checking gesture” which surprised her as she is usually a confident woman.
Adding of the Queen, she said: “She looked positively girlish. When she smiles you can see the young princess coming out. She’s got a beautiful childlike smile and her little hands were formed into fists at one point.
“Perhaps Meghan did bring a little bit more of that out in her.
“We saw displays which showed the pair genuinely having fun.”
Ahead of their engagement, the Queen and Meghan arrived by royal train before travelling by car to the bridge, a £1.86bn project which should ease traffic between Runcorn and Widnes.
They were greeted by local dignitaries as the streets around Runcorn station was lined by schoolchildren and well-wishers who waved flags and applauded as they arrived.
Grange Hill, Hollyoaks, and Brookside creator Phil Redmond, whose wife Alexis is the High Sheriff of Cheshire, was also on hand to greet them.
He joked he should have asked Meghan if she’d like a role in Hollyoaks, adding: “I didn’t ask her, but there’s still lunch yet. She’ll have to go through the audition process like everybody else.”
His wife said: “It’s fantastic to have them here and it’s a big day for Cheshire.”
Mick Noone, former operation director for the council, said the Queen asked how long the six-lane bridge had taken to build, while Meghan described it as “magnificent”.
Queen Elizabeth wore a striking green outfit and hat for the event, as the country remembered the victims and survivors of the Grenfell Tower fire by going green for Grenfell.
Buckingham Palace would not confirm that her colour choice was a show of solidarity with the survivors.
Her outfit was by Stewart Parvin and hat was by Rachel Trevor Morgan. She carried her trademark black Launer handbag and opted for trusty black patent Anello & Davide low heels.
Meghan was wearing Givenchy, the designer label behind her wedding gown.
She wore a cream dress with a cape, adding a thin black belt around her waist. She carried a clutch bag, and opted to leave her hair down instead of wearing a hat. She also towered over the Queen in Sarah Flint stilettos.
Her diamond and pearl earrings were a gift from the Queen, Buckingham Palace confirmed, but it isn’t known exactly when she was given them.
Although the Queen wore gloves, appearing to be neutral to match her granddaughter-in-law’s dress, Meghan did not.
The royal pair took part in a minute’s silence to remember the Grenfell victims at noon.
After the unveiling and ribbon cutting, Meghan and the Queen watched a performance by local schoolchildren called Bringing Communities Together, by performers from the Andrew Curphey Theatre Company.
They appeared to be sharing a joke as they sat next to each other, and Mr Curphey said Meghan was “really calm considering it was her first official engagement with the boss”.
He added: “The children have been hugely excited and it’s been a massive treat for them.
“The Queen said she thoroughly enjoyed it and it was lovely to see the history of the bridge on the videos.
“As Meghan walked away from me she turned back and said ‘the children did great by the way’.”
The Queen and the Duchess were presented with flowers by Savannah Cafferty, 11, and Alarni Wain, who is celebrating her eighth birthday. She said both royals wish her a happy birthday.
As they went from one part of the county to another, there appeared to be some confusion over which car door Meghan should use, and whether the Queen should get into the car first or second.
The royal pair then went on to Chester, 20 miles away, for further engagements.
They attended the official opening of Storyhouse, a library, cinema and theatre complex, where they had a tour of the building.
They met a Syrian settlement group engaged in craft work, and watched a performance by Fallen Angels, a dance theatre company for people who are recovering from addiction.
In the theatre, they saw a medley of songs from Storyhouse’s latest production, A Little Night Music.
After their tour, they greeted the hundreds who had packed the streets outside, as they walked to the City Council, meeting with flag-waving royal fans, some of which went for an American flag in a nod to Meghan’s heritage.
She and the Queen spoke to several people during their walkabout and Meghan was spotted showing her wedding ring to one group.
They collected dozens of floral bouquets and gifts.
She told one person that Harry was the “best husband” and said married life was “wonderful”.
In her second balcony moment with the Queen, Meghan appeared on the Chester Town Hall terrace, alongside Lord Mayor of Chester, Councillor Alex Black, to deafening cheers from the crowds below.
The Queen and Meghan had lunch at Chester Town Hall, as guests of the City Council, concluding their visit.
Meghan married Prince Harry in Windsor in May and since then, has joined the royal family for Trooping the Colour, also known as the Queen’s birthday parade, and has announced her first royal tour.
The Duke and Duchess of Sussex are to make an official visit to Australia, New Zealand, Fiji and the Kingdom of Tonga in the autumn.