Runner-up in 2015 and 2016, Messi's Argentina was looking to banish its past demons in pursuit of a first major international title since 1993.
However, Argentina was upstaged by Colombia, which made a winning start in Group B thanks to Roger Martinez's 71st-minute stunner and Duvan Zapata's late sealer.
As Argentina pressed and probed, first-half substitute Martinez sent a thunderous strike past Franco Armani in Salvador and Zapata made sure of the win during the closing stages.
Argentina came into the tournament off a 5-1 rout of Nicaragua and with the most Copa America victories in history (119) but the South American giant could not find a way past Colombia.
Martinez, a replacement for Luis Muriel in the 14th minute, linked up with Radamel Falcao but his shot deflected wide of the post as Colombia appeared the more likely to strike first.
Overall, it was a forgettable opening 45 minutes, which did not see a single shot on target from either team, and with Messi subdued.
It was more of the same in the second half, until Colombia goalkeeper David Ospina was finally called into action after a vicious volley from Leandro Paredes approaching the hour mark.
The match came to life moments later when Juan Cuadrado's tackle on Messi threatened to spark a melee as Argentina players rushed in to confront the Colombia winger.
That triggered Argentina, which almost took the lead after Nicolas Otamendi's powerful header forced Ospina into a low diving save, while Messi was unable to head home the rebound with the goal at his mercy.
Colombia broke the deadlock with 19 minutes remaining, Martinez collecting the ball on the flank from a raking James Rodriguez cross-field pass, drifting inside before beating Armani from distance.
Fellow substitute Zapata doubled the lead four minutes from the end courtesy of an emphatic close-range finish.
Argentina has work to do before it meets Paraguay on Thursday (AEST) if it is to silence the doubters.
Colombia, meanwhile, will be buoyant and next faces Asian champion and 2022 FIFA World Cup host Qatar, one of the two guest nations alongside Japan at this year's tournament.