The pair were rehearsing together for the first time when Irglova challenged his use of poetic licence in an autobiographical lyric. "She said, 'Why are you bending the truth, when the truth is probably more interesting?'" he recalls.
"I remember realising then this girl is not only smart but she's somebody who's not afraid to face up to the fact that maybe you're lying to yourself. It was very powerful."
About four years later, this conversation formed the basis for the opening exchange between the lead characters in Once, a micro-budget feature filmed in three weeks on the streets of Dublin by Hansard's friend John Carney, who cast his leads after seeing them play a gig together.
At the time, Irglova was 17 and still attending school in the Czech Republic, while Hansard had an established music career, fronting folk rock band The Frames.
He had already played a musician on screen in The Commitments, Alan Parker's 1991 film about an Irish soul band.
However, he says: "I don't remember having any real connection with the creative process during The Commitments other than showing up on set at the right time and speaking my lines.
"The emotional investment was much higher in Once. It's just you and your mates and if anyone's in bad form it's not going to work, so it was a case of throwing yourself into it completely."
Irglova adds: "(Once] never felt like a big film shoot, it always felt like three friends making something together. I have a feeling that was a very special and rare thing that happened."
The stars must have been aligned, because Once went on to surpass all expectations at the box office. Its simple story, charting a developing relationship between two musicians, struck a chord with the public, who responded to the natural chemistry between Hansard and Irglova and the original music they performed in the film.
A couple of pinch-me moments ensued, when the pair won the Oscar for Best Song at the Academy Awards last year and – arguably more prestigiously – were immortalised in an episode of The Simpsons.
There were personal repercussions for Hansard and Irglova too, as life imitated art and their off-screen relationship bloomed.
"You don't go through an experience like that and not come out the other side either hating each other or much closer," comments Hansard, "and me and Mar came out much, much closer."
"The friendship and connection that you're watching on screen between Glen and I exists, and it did grow into a deeper love afterwards," says Irglova.
"Somewhere along the line it became okay for us to become something more than friends and I guess there's a very good possibility it would have never happened had it not been for the film. We're not lovers anymore," she continues, "but we're still very close. We were friends for so long before we became a couple that I don't think it's a connection that will ever be broken."
The pair first met when Irglova's father was promoting a Frames tour of the Czech Republic. Irglova had been classically trained on piano from an early age but also brought up on her father's Leonard Cohen and Joni Mitchell records. Hansard spotted her potential as a singer and songwriter and invited her to contribute backing vocals at some of his gigs.
The cultural exchange continued when Irglova made her first trip to Dublin, staying with the Hansard family when she was in her mid-teens. "Immediately, I was welcomed, they just put their arms around me and put a cup of tea in front of me."
She fell in love with the city then, and still lives just outside Dublin. These days, she speaks perfect English with a pronounced Irish accent. The Czech inflections, so charming in Once, are fighting a losing battle. Her songwriting, however, has flourished.
"When I met Glen, he said, 'Just play along to my songs' and that was completely new for me," she says. "I would listen to the song and I would hear a melody in my head and that's still how I do it to this day. Whatever I hear in my head is what I play. I think Glen is to blame for this, really."
According to Hansard, it's a symbiotic relationship: "I didn't discover it immediately, but after a couple of years of playing music with Mar, I realised I had found a musical partnership that not only matched my relationship with my own band but surpassed it in some ways too."
The latest result of that partnership is the new Swell Season album, Strict Joy. Perhaps as an inevitable consequence of their romantic split, the 12 songs on the album were written separately, with the lion's share of the content coming from Hansard.
"It would be very easy to say this record is about my relationship with Mar and to a large extent it is, but it would also be untrue to say that it's exclusively about Mar," he says, describing it as an "audio diary".
"What I find most interesting about diaries is that the happy day is always blank. There's an idea that happiness is quiet and sadness is loud because stuff is created out of sadness. I guess what I'm saying is that when you need it, your instrument is often medicinal.
"You pick up your diary or your guitar when you are feeling miserable or reflective or when you're in that half-mood which I guess has a melancholic atmosphere."
The Swell Season will spend the rest of the year touring the US, landing in Scotland in time for Celtic Connections in January. Hansard sounds almost relieved to be back in his natural environment after his brush with cinematic notoriety.
"What do you do with that much success?" he asks, before providing his answer: "I think ultimately what you do is take a holiday, drink a bottle of champagne, get some perspective and wake up one morning, clap your hands together and say, 'OK, back to work'."
Irglova is uncertain whether she would want to repeat her acting experience: "With film, it's all about the actor being able to feel the things that the character's feeling. It must do some strange things to your mind. Music I find much easier because you're being honest about where you are as a person."
There it is, that fidelity to the truth thing again. But should she ever want to revisit her character from Once, Hansard and Carney have already worked out what happens in the sequel.
"The guy goes to London and ends up working on a building site with a bunch of Irish lads and becomes an alcoholic," says Hansard.
"She comes over a few years later trying to find him, and he's drunk in a bar in Kilburn. She says, 'Do you not remember me?' and he's like, 'Who are you? … Oh, yeah, did you get that piano I bought you?' That would be the opening line."
Strict Joy is out on Monday on Anti-Records. The Swell Season play the City Halls, Glasgow on 16 January as part of Celtic Connections