We caught up with the couple on "TMZ Live" Monday, where Brantley and Amber shared the surprising specifics surrounding their 3rd child's arrival into the world.

As we previously reported, Amber found herself giving birth in Brantley's tour bus last Friday, as her country star hubby was performing a show in Tupelo, Mississippi.

As Brantley put it ... he initially thought there was a security threat when he was suddenly pulled offstage mid-show. However, he was soon informed of his wife's situation ... and Brantley is praising his lady as "the real rockstar" of their family.

Brantley certainly wasn't wrong ... watch the video, as Amber goes into detail about how she sprung into action when she unexpectedly went into labor with no time to make it to a hospital.

Yet, giving birth on the tour bus wasn't her only impressive moment from the evening ... she notably gave Brantley her blessing to get back up onstage once baby boy Gilbert made his arrival.

Brantley added ... "She looked at me with this understanding and said, 'You need to go finish.'"

Though, according to Brantley, he did get to hold his baby for a bit before returning to his gig ... which allowed Amber to clean herself up a bit before going into mom mode. Watch the video -- they had even previously joked about a "Lion King" reveal.

In addition to their new arrival, Brantley and Amber are parents to 6-year-old Barrett and 5-year-old Braylen.