The two nations are set to meet in Chile, drawn together in Group C of the continental competition - and will go head-to-head in Valparaiso on June 18.
Their past four outings had seen 17 goals, and it was only a matter of time before the opening goal came - with Josef Martinez providing it in the 60th minute, and it proved to be the winner.
Martinez's third international goal broke the deadlock, the Torino striker storming into the penalty area and tucking away a powerful finish from the right-hand side of goal.
The two chances were indicative of the end-to-end nature of the contest in Fort Lauderdale, which saw plenty of aggression towards both goal and opponent.
Deza's strike deflected off a Venezuelan defender and into the arms of goalkeeper Alain Baroja, with the six-time Peru international seemingly picking up an injury in the play - and was replaced by Christian Cueva moments later.
Emotions were running high after the break too, with an all-in eventuating in the 53rd minute - which saw each of Roberto Rosales (Venezuela) and Andre Carrillo (Peru) carded.
Moments earlier, Venezuela's Salomon Rondon had the chance to open the scoring but, after taking two touches inside the area, the Zenit striker could only poke a shot over the crossbar.
Cesar Gonzalez's acrobatic effort nearly doubled Venezuela's lead, the midfielder throwing himself at a bouncing ball into the area - but his fly-kick went wide of the target.
Some clutch one-on-one goalkeeping from Baroja and Peru goalkeeper Pedro Gallese frustrated the two nations' respective attackers late on, but ultimately, Noel Sanvincente's men took the spoils ahead of the Copa America.
