Micah Richards fired his spot-kick over the crossbar in sudden death as City failed to overhaul Brendan Rodgers' Reds at the top of Group B.
Stevan Jovetic cut in off the left flank to open the scoring in the 35th minute at TCF Bank Stadium in Minneapolis - his fifth goal in three games in this tournament - but City's lead lasted just two minutes as Greece Under-21 striker Dimitrios Diamantakos levelled up the scoreline after a slick move.
After David Silva was brought down in the box, an Aleksandar Kolarov penalty in the 53rd minute restored City's lead.
But Diamantakos scored again in the 66th minute, rounding Joe Hart who had raced off his line, and the match went to penalties.
Silva and Kolarov stuck away their spot-kicks but James Milner and Scott Sinclair both failed with theirs.
After Bruno Zuculini had scored, Hart made a superb save with an outstretched leg from Avraam Papadopoulos to keep City alive in the shootout and take it to sudden death.
John Guidetti then found the net, but Richards was ultimately the fall guy, blazing his effort over, which allowed Thanasis Papazoglou to seal victory for the Greeks.
City now return home for the FA Community Shield clash against Arsenal at Wembley on Sunday, August 10.