Due to the game being played outside of FIFA dates, Argentina coach Diego Maradona chose to select players exclusively from the Argentinian league, and his decision paid off in the 26th minute as Huracan midfielder Matias Defederico put the hosts ahead in Santa Fe.
Three minutes later, however, Nelson Barahona equalised for Panama when he pounced on a rebound on the edge of the area.
But Bergessio restored Argentina's lead shortly before half-time when he headed in a Daniel Montenegro corner, then six minutes from time he ensured the win when he volleyed home another cross from Montenegro.
Maradona admitted to being "dazzled" by his new squad, and confirmed some of the players may have won the right to remain in the squad for the upcoming World Cup qualifiers at home to Colombia and away to Ecuador.
"I'm very satisfied. This players showed a desire to be part of the national team. They dazzled me," he said.
Bergessio added: "The balance was quite positive, all the squad showed we can wear this shirt as well as the European-based players do."