Dynamo Kiev midfielder Gerson Rodrigues condemned the Republic to its second defeat in as many Group A games with a superb long-range effort five minutes from the end.
"It was a horrible and embarrassing night," Ireland skipper Seamus Coleman said.
The player of the match accolade has never been more meaningless, but Ireland captain Seamus Coleman is as honest as always.— RTÉ Soccer (@RTEsoccer) March 27, 2021
Live now @RTE2 and @RTEplayer #RTESoccer #IRLLUX pic.twitter.com/ItIutuboUA
"We got what we deserved which was nothing. I don't think we demanded the ball enough and looked to break them down as quickly as we could."
Ireland sits bottom of the group on zero points, four behind leaders Serbia and Portugal, who drew 2-2 in Belgrade.
Luxembourg, meanwhile, is a point behind the two leading teams after a rare FIFA World Cup qualifying victory for the tiny country.
The win at Aviva Stadium came thanks to smart play from the 25-year-old Rodrigues, who pounced on a knockdown to lash home his sixth international goal.
Luxembourg has never qualified for a major tournament and has little chance of making it out of the five-team Group G and to next year's FIFA World Cup finals in Qatar.
However, despite hailing from a country with a smaller population than many provincial European cities, Luc Holtz's side inflicted an humiliating defeat on the Republic.
Stephen Kenny is still looking for his first win as Republic of Ireland boss, with his run of games without a win since taking charge last year now at 10.
His next opportunity will come on Wednesday (AEDT) when Ireland faces FIFA World Cup host Qatar in a friendly in the Hungarian city of Debrecen.