City boss Manuel Pellegrini named new signings Fernando and Willy Caballero in his starting line-up against the newly-promoted Scottish Premiership team at Dens Park.
But Brazilian midfielder Fernando, a £12 million capture from Porto, and Argentine goalkeeper Caballero, signed from Malaga for £6 million, were unable to stop lacklustre City falling two goals behind before Pellegrini made nine changes at half-time.
Dundee, promoted as champions of the Scottish second tier last season, took the lead in the 27th minute when Gary Harkins was awarded a spot-kick after he was brought down by Dedryck Boyata.
Harkins saw his penalty saved by Caballero, but the midfielder pounced to slam in the rebound.
Stevan Jovetic came close to netting an equaliser for City shortly afterwards as he shot into the side-netting.
Yet just over 10 minutes later Dundee doubled their lead when Luka Tankulic, one of three trialists for the home side, fired past Caballero via a substantial deflection off Boyata.
Only Caballero and Wales midfielder Emyr Huws remained on the pitch for City after the break as Pellegrini made wholesale changes, hauling off the likes of Jesus Navas, Jack Rodwell and Aleksandar Kolarov.
But apart from a half-chance from Alvaro Negredo and a Samir Nasri effort, City struggled to create any real chances in a tame opening to the build-up to their title defence.
City continue their tour of Scotland with a game against Hearts at Tynecastle on Friday before heading to the United States for friendlies against Sporting Kansas City, AC Milan, Liverpool and Olympiakos.
They face Arsenal in the Community Shield at Wembley on August 10 and kick off the new Premier League season at Newcastle on August 16.