Martinez's stunner was followed by a late Duvan Zapata finish as Carlos Queiroz's side opened their Group B campaign with an impressive victory against an off-colour and disappointing Argentina outfit.
It is the first time that Colombia have beaten La Albiceleste since 2007, but it was far from a fluke result.
There was a Premier League pairing at the heart of the Colombia defence with Everton's Yerry Mina joined by Tottenham Hotspur's Davinson Sanchez, and both needed to be switched on in the early moments as Argentina threatened an early breakthrough in Bahia.
Quieroz's side were forced into a change in the 14th minute, though, with Martinez replacing Muriel as the latter struggled to shake off an injury that he picked up earlier in the contest.
The incoming attacker actually had the first serious chance of the match in the 17th minute, but Martinez's strike was deflected just wide of the Argentina post during a lively period.
Colombia were the team looking the more likely to make the breakthrough in the Group B clash, although the first 30 minutes passed without too many talking points.
Lionel Messi was doing his best to make things happen in the final third. Angel di Maria and Sergio Aguero were both extremely quiet, though, and there was a mistake from the Argentines down the other end with Nicolas Otamendi only just getting away with a loose touch inside his own box in the 31st minute.
James tried his luck for Colombia in the 35th minute after working space in a dangerous position, but the Real Madrid attacker missed Franco Armani's crossbar, before Falcao headed wide of the same goal moments later.
Falcao had another half-chance five minutes before the break after meeting a cross from Juan Cuadrado - Otamendi was in the way, though, and the two teams headed down the tunnel at 0-0. Neither side produced a shot on target in the first 45 minutes, but Colombia had five attempts to Argentina's one.
Rodrigo de Paul had actually taken the place of Di Maria at the interval, which was not a surprise as the Paris Saint-Germain attacker struggled badly in the first half of action.
Messi managed to dance into the Colombia box just before the hour and the stadium waited for a shot - Mina had other ideas, however, and came from nowhere to make a fine challenge.
Argentina were on the front foot, though, and David Ospina had to keep out a powerful drive from German Pezzella, before a late challenge from Cuadrado on Messi saw both sets of players become involved in a pushing match.
Scaloni's team had two excellent chances in the 67th minute through Otamendi and Messi, but the former saw his header kept out by Ospina before the latter somehow nodded wide of the post from close range.
It was Colombia that made the breakthrough in the 71st minute, however, courtesy of a spectacular goal from Martinez, who cut onto his right foot before finding the back of the net with a thumping effort from distance.
Messi tested Ospina's handling from a long-distance free kick in the period that followed, but Colombia had their lead intact entering the final 10 minutes of the Group B affair.
The contest was all but decided in the 86th minute, however, with Duvan Zapata, who scored 28 times for Atalanta BC during the 2018-19 campaign, turning a low Jefferson Lerma cross into the back of the net to stun the Argentine fans inside the stadium.
Lo Celso struck wide of the post late on as La Albiceleste looked for a late response. It was not to be, though, with Colombia opening their 2019 Copa America campaign with an extremely impressive win.
Next up for Argentina - already under early pressure in the competition - is a clash against Paraguay on Wednesday, while Colombia are in action against Qatar on the same day.
ARGENTINA (4-2-3-1): Armani; Saravia, Pezzella, Otamendi, Tagliafico; Paredes, Rodriguez (Pizarro 67'); Di Maria (De Paul 46'), Messi, Lo Celso; Aguero (Suarez 79')
COLOMBIA (4-3-3): Ospina; Medina, Mina, D Sanchez, Tesillo; Cuadrado (Lerma 64'), Barrios, Uribe; James, Falcao (D Zapata 81'), Muriel (Martinez 14')