Town striker James Collins started for the Republic of Ireland as they suffered a 2-0 defeat to Switzerland in their Euro 2020 qualifier on Tuesday night.
Victory would have booked the Irish's place at the finals, but Haris Seferovic and Edimilson Fernandes were on target to secure a Group D victory for the Swiss.
Collins kept his place in the side from Saturday's goalless draw in Georgia, although he was partnered upfront by Brighton's Aaron Connolly, who had a spell on loan at Kenilworth Road last term.
The hosts took the lead on 16 minutes when Borussia Monchengladbach's Breel Embolo set up Seferovic to finish through Shane Duffy's legs from outside the box.
Mick McCarthy's side couldn't create any clear-cut chances for Collins during the opening 45 minutes, as Town's top scorer was replaced by Callum O'Dowda at the interval.
The Republic did rally in the second period, Glenn Whelan seeing his shot blocked, as they only managed two attempts on target.
Everton defender Seamus Coleman was then sent off for his second yellow card after being adjudged to have handled an Embolo shot inside the area with 15 minutes to go, keeper Randolph superbly saving Ricardo Rodriguez's spot-kick.
Fabian Schar also headed against the Irish woodwork in the second half before Fernandes's clinching goal late on, despite the best efforts of defender Shane Duffy to keep it out.
The result means Ireland still top the group, but now need to beat Denmark on November 18 to ensure qualification.
McCarthy said: "We've still got a chance to qualify automatically.
"It's a cup final in the Aviva (Stadium in Dublin) and I'd have taken that.
"The unlikely thing was that we were going to come here and beat that Switzerland team. We'd have had to have played to our maximum.
"We're playing Denmark and I'd have taken that at the start and I'll take that now."
"Switzerland proved they were a very good side.
"No shock to me. They have good players, who have played together for a long time.
"They were dominant in the first half but we had a second half we can be proud of.
"We haven't scored that many goals. In the past we have managed to keep a clean sheet but we didn't this time."
'We can get the three points against Denmark."
Meanwhile, Connolly, who made two substitute appearances for the Hatters last term, and was handed his full Ireland debut on Tuesday, added: "It was a disappointing night but there is still all to play for.
"I believe we can play against Denmark and get three points and qualify."