"When Saturday at 11:11, they showed us the first baby that was crying, we were crying," Angelil told Canadian entertainment show etalk on Monday.
Dion, 42, has been open about her fertility struggles and conceived the twins on her sixth in vitro attempt over the course of about a year.
"It was a very emotional moment, and even though we didn't speak to each other, we were both thinking about the time it took," said Angelil. "We are very lucky, very privileged and very fortunate to have these boys."
The birth wasn't just a dream come true for the couple, but also for their 9-year-old son, René-Charles.
"He was very, very happy that it was boys," Angelil said on Monday. "I remember a few months ago when mom told him the day we were going to know if it was a boy or a girl or two boys or two girls ... his reaction pointing to his mom's belly he said, 'I hope there's not too many girls in there.' "
The fraternal twins, whose names have not been released, have dark hair and distinctly different looks. According to Angelil, René-Charles observed, "One looks like dad, and the other one looks like mom."
Although Dion relishes her role as mom, she is also looking forward to her next big project: returning to the Las Vegas stage.
"We know she's going to start rehearsing Jan. 17," says her husband. "It gives her three months to recoup and she thinks she can do it."