Martinez made up for the miss when he converted a low cross from De Paul 17 minutes into the second half.
The win, combined with Brazil’s 0-0 draw at Colombia earlier in the day, narrows the gap between leaders Brazil and second-placed Argentina to six points at the top of the 10-team South American qualifying group.
The top four teams qualify automatically for Qatar and the fifth-placed side go into an inter-regional playoff.
Uruguay, who ended the day fourth in the 10-team table, were unbeaten in eight matches coming into Sunday’s match.
They took the game to their rivals, with Luis Suarez coming close on three occasions in the first half hour, including one thunderous shot that came back off the post.
Argentina grew as the game went on, though, and Giovani Lo Celso missed a golden chance to open the scoring when he rounded the goalkeeper but delayed before striking a shot that hit the underside of the bar and the goal line before rolling to safety.
After the home side drew first blood they never looked like losing an unbeaten run that now stands at 24 games.
Edinson Cavani came on in the second half for Uruguay but it was Argentina who had the better chances in an entertaining game that featured only 15 fouls and no cautions.
Messi came close to getting his second and substitute Angel Di Maria could have made it four on two occasions only to twice see Muslera parry his goalbound shots.