Air carrier at FL380 with Bogota Control experienced potential conflict with opposite direction traffic at the same altitude; reporter indicating TCAS traffic advisory prompted the flight crew to turn away from the conflict prior to any recognition action on the part of ATC.

Narrative

FL380; cruise mach with Bogota Control; LNAV and VNAV engaged; precisely on published airway; no offset. Between points PUKEN and IROTI; northbound on the airway. VHF quality with Bogota was loud and clear. One TCAS target on view; 4000 FT below us; ten miles on front of us; also northbound; provided confirmation that TCAS was functioning normally. New TCAS target appeared 21 miles north of us; at the same altitude (FL380; TCAS indicating 00 relative altitude). This new target was southbound; also on the airway. Captain flying; Captain and relief pilot remarked to each other that the target is southbound and co-altitude. Captain transmitted; 'Bogota; do you have traffic at FL380?' Bogota responded with several excited sentences in Spanish. Captain simultaneously used heading select to turn right 30 degrees or so to about 030. Bogota then gave us a heading of 040. The other aircraft passed off our left wing at approximately 5 miles; in the Captain's estimation. The TCAS audible warning; as the two aircraft passed each other; was only 'traffic' and amber color; due to the offset achieved by the pilots reacting prior to the Controller responding. Cause of conflict is unknown; but it appears to be similar to the mid-air over Brazil; the difference being a fully functioning TCAS and good weather at night allowing pilots to prevent a collision.