Martinez netted his 19th international goal from Marcos Acuna's cross as already-qualified Argentina moved to 35 points from 15 qualifiers, with 13 wins from their past 15 games.
La Albiceleste were missing talisman Lionel Messi but welcomed head coach Lionel Scaloni back to the touchline after COVID-19 and he would have been pleased with his side which was largely in control throughout.
Colombia, whose stuttering qualifying campaign is in strife after seven games without a win nor a goal, struggled to retain possession and mustered few genuine chances on goal against a stubborn Argentine defence, who have kept five clean sheets from their past five home qualifiers.
Messi's Paris Saint-Germain teammate Angel Di Maria, who was dangerous throughout, curled a 16th-minute effort just wide of Colombia goalkeeper Camilo Vargas' goal.
In the 29th minute, Vargas could not prevent Argentina going ahead when Inter forward Martinez trapped Acuna's cross inside the box and lashed home, squirming under the goalkeeper's grasp.
⚽️⚽️GOAL⚽️⚽️ A predator's finish from Lautaro Martinez to give @Argentina the lead over @FCFSeleccionCol!— beIN SPORTS (@beINSPORTS_AUS) February 2, 2022
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Out of nowhere, Colombia almost levelled on the stroke of half-time, with Luis Diaz setting up Miguel Borja who was denied by Emiliano Martinez. The new Liverpool signing's follow-up effort was cleared off the line by German Pezzella.
Acuna forced a good save from Vargas early in the second half, while the Colombian goalkeeper made a better block with his outstretched right arm to deny Di Maria from range in the 65th minute.
Giovani Lo Celso, who left Tottenham for Villarreal on loan on deadline day, flashed a chance over the bar as Argentina survived a late scare from Johan Mojico's shot which almost caused a Nicolas Gonzalez own goal, although that would not have counted due to offside.
Argentina should have grabbed a second in the 94th minute when Gonzalez broke in behind but opted to square for Paulo Dybala allowing the Colombian defence to clear.