The play-off semi-final in Bratislava ended goalless after extra-time and in the shoot-out Browne saw his penalty saved by Slovakia goalkeeper Marek Rodak.
Tottenham's Matt Doherty hit the bar with his spot kick as the Irish bowed out 4-2 on penalties.
Browne came on as a 60th minute substitute and made a real impact in an attacking midfield role.
He forced a good diving save out of Rodek within 10 minutes of coming on and then in extra-time saw an effort come back off the post.
The Republic were the better side but couldn't make that count, the home side then holding their nerve from the spot in the shoot-out.
Slovakia will play Northern Ireland next month in the play-off final.
Browne has two more games during the international break, with the Republic facing Wales and Finland in the Nations League.