The second-youngest U.S. team ever fielded -- with an average age of 22 years, 154 days -- started brightly but found itself trailing inside of 10 minutes when Jose Fajardo's glancing header from an Alejandro Yearwood cross beat Zack Steffen and gave Panama the early lead.
The Americans got back on level terms 10 minutes later, when Reyna scored from a free kick at the top of the penalty area after Yunus Musah was brought down following a nice run at the Panama defense.
The goal was the first in Reyna's U.S. career, just days after his 18th birthday and a reported new deal with German club Borussia Dortmund.
Gioacchini then fired the U.S. in front with his first international goal, pouncing on a rebound from Ulysses Llanez's shot after some sloppy play out of the back from Panama.
Firmly on the front foot, the U.S. made it 3-1 in the 26th minute, with Gioacchini -- who plays at Ligue 2 side Caen -- netting his second with a close-range header following some neat buildup play in the penalty area.
Following the match, Reyna said of Gioacchini's goals: "I think Nico did what he was supposed to do. You saw in the first game, maybe we lacked a striker's presence in the box. Well done to him. We're happy for him.
"It was good to end this camp like that. Looking forward to getting back with these guys again soon."
Panama came out of half-time trailing by two goals and looked dangerous in the opening 15 minutes of the second period but were unable to find another goal.
Gioacchini had a chance to make it a hat trick on the night, but he missed from the penalty spot after Panama's Oscar Linton was whistled for hand ball in the area.
Panama clawed one back before the 80-minute mark in a bizarre sequence that saw Fajardo break free in the U.S. area with only Steffen to beat, and he sliced a left-footed volley into the net to make it 3-2.
Substitutes Soto and Richie Ledezma put the game away soon after the Panama goal, with the former meeting the latter's cross in the box and heading home for his first tally in the U.S. senior side.
Sebastian Lletget made it 5-2 before full-time with a header of his own before Soto headed in his second from Ledezma to reach the final scoreline.
Midfielder Yunus Musah became the first American male to start multiple games before the age of 18 after starting in his debut vs. Wales. Musah turns 18 on Nov. 29.
"Really happy with him in camp, the guys took to him well," U.S. manager Gregg Berhalter said of Musah, who has yet to officially decide where he will play internationally between the U.S., England and Ghana. "All we want to do is create a good environment that [players] can trust, that's good for their development.
"In the end, it's gonna be him and his family who decide [whom he represents]. We'll be here to answer questions, but it'll be him who decides."
Berhalter also singled out Gioacchini for praise after the match.
"He's a fantastic kid. I've had a lot of conversations with him leading up to this," the U.S. manager said. "You could tell he was a little bit apprehensive. My job was to tell him that he's good enough, give him confidence. He gave us what I think we were missing a little bit against Wales.
"It's good to have him. A shame he didn't get the hat trick in his debut start, but a great performance for sure."
On Sunday, Berhalter hinted that the team is looking to schedule one more friendly before the end of the year, likely to be played on home soil.