The loss puts Argentina in an early hole at the tournament where they have finished runners-up in four the past five editions. For Messi, it adds more pressure as the Barcelona star tries to win his first senior trophy at the international level.
Martinez's stunning strike came in the 71st minute and Martinez's fellow substitute Duvan Zapata made sure Colombia took all the points by pouncing from close range to knock in a cross from Jefferson Lerma 15 minutes later.
"This is not the way we wanted to begin," Messi said after the match. "There was a little bit of nerves, of wanting to have a good debut, of not wanting to commit mistakes."
Colombia had looked stronger in the first half but Argentina were a different beast when Rodrigo de Paul replaced Angel Di Maria immediately after the interval and Messi was able to find space to cause some problems.
Messi wasted the best chance of the game before Martinez's piledriver broke the deadlock when he skewed a rebound wide from close range, and also went on a thrilling solo run into the area but could not get a shot away.
A goal from Lionel Scaloni's side looked imminent but Colombia's usual talisman James Rodriguez split their defence with an impressive long pass to Martinez, who came on early in the game after Luis Muriel was forced off injured.
Martinez's stunning strike floored Argentina and although they came back with a couple more attempts, they never looked like scoring.
"We have to keep our heads high and think about what's next, because this is just the beginning and there's still a lot of Copa to be played," Messi said.
Colombia captain Falcao had high praise for opponent Messi after the match.
"[Argentina] is always a very difficult opponent and they have the best player in the world," Falcao said. "We had the challenge of beating them for the first time after a long time. We're in a maturation process and we were close, and we knew this was a good opportunity."
Colombia top Group B with three points ahead of Qatar's meeting with Paraguay on Sunday.
Information from Reuters was used in this report.