Ramos beat the offside trap to run onto a through pass from winger Fredy Guarin, round goalkeeper Leonel Moreira and put the Colombians ahead seconds before half-time.
Costa Rica had been reduced to 10 men after 27 minutes when forward Randall Brenes was shown a straight red card after a dangerous tackle on defender Luis Perea.
Colombia lead the group with three points ahead of hosts Argentina and Bolivia who have one point each after their 1-1 draw on Friday in the tournament's opening match in La Plata.
"The important thing was to start with a win and that's what we did," Ramos told reporters.
Colombia, who face Argentina on Wednesday in Santa Fe, dominated after a bright start by Costa Rica but their passing was sloppy in a game mostly fought out in midfield.
They had their first clear opportunity in the 12th minute when Radamel Falcao headed down a cross and Guarin hit a powerful shot that Moreira pushed away for a corner.
Their task was made easier when Brenes, one of the more experienced players in a young Costa Rica side, was dismissed and Ramos took advantage to score just before the break.
Substitute striker Hugo Rodallega had a chance to extend Colombia's lead in the 54th but Moreira made another fine save.
The goalkeeper, asked if his performance had prevented a heavier defeat, said: "This is my job and I have my team mates' back-up... we're all fighting together and it's not just the goalkeeper."
Moreira added that the loss complicated matters for Costa Rica but said the Central American team now had to just focus on their next match against Bolivia in Jujuy on Thursday.
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