Jose Cancela, Yosimar Arias, Pablo Salazar and Jose Cordero also scored for the Costa Rican side, which eclipsed the record for most goals in a Champions League game set in Pachuca's 7-1 victory over Jalapa and matched by Toluca in a 7-0 win over Marathon, both in 2009.
Herediano dominated from the opening kickoff with the Guyanese amateurs struggling to generate any possession, and even less in the Herediano half.
Herediano, which was making its second appearance in the Champions League and first since 2009, created chances as early as the sixth minute, forcing Alpha keeper Ronson Williams to tip a shot over the bar.
Williams blocked several shots with athletic saves, most notably in the 40th, 43rd and 58th minutes, but with Herediano controlling nearly twice the possession as Alpha, he was simply overwhelmed.
Alpha United will get its home leg on August 4, when it returns to Providence to host Herediano.
The Costa Rican side went ahead from a short corner, which was floated to the back post by Marvin Obando. Vargas headed the ball, which bounced down off the crossbar and crossed the line, and Jose Cubero made sure by slamming in the rebound.
It took Alpha nearly 12 minutes to hold any possession in the Herediano half, and 19 minutes before it managed a shot - a speculative effort from 40 meters, but also had two dangerous moments when Kirk Duckworth shot just wide of the right post from inside the penalty arc in the 27th and another in the 45th also narrowly missed.
But Herediano doubled its head when Cancela pounced on the rebound of Anderson Andrade's shot, initially blocked by Williams, and Yosimar Arias hike the advantage to 3-0 in the 38th when he volleyed Andrade's cross from 14 meters.
Salazar tacked on his goal in the 59th, Barbosa, who replaced Andrade in the 67th, added three more in 70th, 77th and 89th minutes before Cordero finished it with a tally in the fourth minute of injury time.
Herediano, which was playing its first competitive match since April, is set to open its Invierno season on Sunday against Puntarenas.
The match marked the debut of Alpha United in the Champions League, having qualified as the third-place finisher in the Caribbean club championship.