Alexander Isak and Florian Wirtz put the Reds ahead, but like against Leeds last week in Chicago - when Liverpool were two goals up but lost 3-2 - they were unable to preserve their lead.
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Aleksandr Golovin's penalty and Mika Biereth got Monaco back on level terms, before Paris Brunner headed in a late winner.
Iraola made 11 changes to his side in the second half and gave a first Liverpool appearance to new signing Victor Munoz.