Martinez's instinctive 10th-minute finish was his sixth goal in 10 international appearances and due reward as Lionel Scaloni's men made a rapid start.
They faded during the second half but Lo Celso capitalised upon an error from Venezuela goalkeeper Wuilker Farinez to settle matters at the Maracana - the stadium where Argentina lost the 2014 World Cup final to Germany.
It means Lionel Messi's ambitions of a first international title remain on track, with Argentina's bitterest rivals now standing in his way.
Martinez displayed his burgeoning understanding with Sergio Aguero in attack by playing in the Manchester City striker with a deft early pass – Farinez saving with his feet.
The set-piece deliveries of Messi and Leandro Paredes were causing Venezuela problems and, after an unmarked German Pezzella should have opened the scoring, Martinez duly did.
The Inter star converted a wonderful backheel from Aguero's miscued shot and quickfire bookings for captain Tomas Rincon and fellow midfielder Yangel Herrera soon afterwards suggested Venezuela were losing their composure.
Jhon Chancellor powered over a chance to equalise in the 40th minute, although Rafael Dudamel's side needed a brilliant clearance from Roberto Rosales to prevent Martinez from turning in a second after slick interplay between Messi and Marcos Acuna.
Martinez crashed against the back stanchion after Paredes unpicked Venezuela with a sumptuous 48th-minute pass.
Dudamel boldly sent on winger Yeferson Soteldo for centre-back Luis Del Pino and Argentina were forced to absorb sustained pressure for the first time in the contest.
Franco Armani saved sharply from Ronaldo Hernandez after Scaloni's decision to withdraw Martinez and Acuna raised eyebrows.
Farinez's work at the other end contrasted horribly when he spilled Aguero's 74th-minute shot to leave Lo Celso with a tap-in.
What does it mean: Scaloni's Argentina taking shape before showdown with old enemy
Scaloni's sparse coaching credentials were pored over as Argentina staggered through the opening matches of their campaign. This was easily his team's best performance of the tournament until his questionable second-half changes, with Paredes excelling in a midfield three where Acuna provided much-needed edge. La Albiceleste's attacking prowess has never been in doubt but a shaky backline enjoyed greater protection and Armani handled soundly throughout.
Martinez shows ruthless streak again
As he did in the vital group win over Qatar, Martinez gave Argentina an early lead – his supreme poaching instincts to the fore. More importantly, the 21-year-old appears entirely unhindered by the weight of expectation and failures that has sapped many of his compatriots over recent years.
Messi short of brilliant best
It was largely a game of personal frustration for Messi, who was largely well shackled by combative opponents. But his delight at Argentina's progress was clear and it bodes well that a team so shambolic of late were able to prevail with their talisman short of the A-game he will surely need against Brazil.
What's next: The Copa clash the world wants to see
As unlikely as it seemed when Argentina toyed with group-stage elimination, the two great rivals in South American football now have the chance to produce a clash for the ages in Belo Horizonte on Tuesday. Despite their shot-shy efforts against Paraguay, Brazil will be favourites. But are Argentina just about clicking at the right time?