The duo - consisting of John Paul White and Joy Williams - toured with the 'Rumour Has It' star last year and revealed that everybody in the Staples Centre in Los Angeles on Sunday (February 12) was anticipating her performance.
Speaking about Adele's six Grammy wins, Williams told Digital Spy: "It was no surprise for any of us that that was going to be a remarkable night for Adele.
"Nobody was cheering more loudly than John Paul and I were for her. Everybody was rooting for her and it was lovely to be able to catch up backstage."
Talking about Adele's performance of 'Rolling In The Deep', White added: "I think that's what everyone was waiting for; to hear her.
"The beauty of her standing on stage with just her band and a total lack of bells and whistles and production really highlighted just how great a singer she really is.
"If the music is great you don't need it. It's all about the ears and not the eyes to paraphrase her."
The Civil Wars also walked away with two Grammy Awards - 'Best Country Duo/Group Performance' and 'Best Folk Album' for their debut record Barton Hollow.
Commenting on the wins, White explained: "It feels like it was an out-of-body experience. I keep wondering if it actually happened as I'm back home emptying the trash and cleaning the bathroom!"
Williams added: "I think it might hit us a little bit more when they arrive. You get the Grammy handed to you, but they've not actually been inscribed.
"So the win is a total high and then as soon as you turn the corner it's taken off ready to be given to the next presenter. We're not complaining and we're glad to have a few coming in the mail soon!"
The Civil Wars will return to the UK in March for a headline tour, which will start at Birmingham's HMV Institute. Tickets for the shows are on sale now.
Watch the music video for The Civil Wars' hit 'Barton Hollow' below: