The result meant the Irish fell short in their bid to qualify automatically for Euro 2020 and they will have to negotiate a play-off in March.
Browne was named in the starting XI after impressing Mick McCarthy in last week's friendly against New Zealand.
He played the full game, while Maguire joined the action from the bench in the 82nd minute.
Martin Braithwaite gave Denmark the lead in the 73rd minute, Wolves' Matt Doherty equalising five minutes from time.
Despite late pressure, McCarthy's side couldn't find the second goal needed to send them to the Euros automatically.