Argentina did not impress but the extra man told in the final quarter when substitutes Joaquin Correa and Angel Correa broke through.
Venezuela did not go home empty handed, though, as Yeferson Soteldo scored a cheeky “Panenka” penalty deep into injury time at the end of the game.
Ahead of leaders Brazil’s trip to Chile, Argentina are three points off the top of the single CONMEBOL qualifying group.
The two South American giants meet in Sao Paulo on Sunday.
Argentina had been marginally the better side early on, hitting the bar through Rodrigo de Paul’s long range strike on 13 minutes.
But the match turned decisively in the visitors’ favor on the half hour as Adrian Martinez, who was an early substitute and had been on the field for less than five minutes, lunged in rashly on Messi.
The referee brandished a yellow card but following a VAR review that was upgraded to a straight red card for the ugly studs up challenge.
The match turned into attack versus defense and Argentina secured the crucial breakthrough in first half injury time.
Messi was given a great chance to get his name on the scoresheet five minutes from time but shot too close to Farinez
As the pressure mounted, Giovani Lo Celso slipped in a perfectly timed pass for Lautaro Martinez to slot into the bottom corner.
Argentina laboured after the break against a packed defense but substitute Joaquin Correa finally put the match to bed 20 minutes from time with a fierce low strike from just inside the area after good work from Messi and Lautaro Martinez.
Three minutes later, Lautaro Martinez’s snapshot was stopped by Venezuela goalkeeper Wuilker Farinez but Angel Correa mopped up the rebound.
Messi was given a great chance to get his name on the scoresheet five minutes from time but shot too close to Farinez.
Substitute Alejandro Gomez then stood on Soteldo’s foot in the penalty area, allowing the diminutive playmaker to salvage some Venezuelan pride from the spot.
Ecuador left it late to cement their hold on third place in the standings before beating Paraguay 2-0 in Quito.
Felix Torres opened the scoring two minutes from time with a header and Michael Estrada’s individual effort sealed the victory five minutes into time added on.
Fernando Saucedo equalised seven minutes from time for Bolivia in a 1-1 draw in La Paz after Roger Martinez’s opener for Colombia on 69 minutes following a mazy run.