Mario Hermoso opened the scoring, before Castro converted a penalty and Koulierakis marked his Roma debut with a goal of his own, with Matias Soule completing the rout in the second half.
Vice president Ryan Friedkin and new CEO John Galantic were both present to watch Gian Piero Gasperini's side ease to victory over the Welsh fourth-tier side, though tempers flared briefly in the second half after a heavy challenge on Paulo Dybala, sparking a brief confrontation between the two benches before things settled down.
Hermoso, deployed by Gasperini in an unfamiliar central midfield role, was one of the standout performers, scoring the opener and winning the penalty that Castro converted.
"We played a great game, especially in the second half," Hermoso said afterwards. "They started to press and it was a bit harder, but we started to play.
"For me it was a bit of a different role, but I felt good with my team-mates. The important thing is winning, because that gives us confidence for the next matches too."
Asked how Gasperini had explained the positional switch, Hermoso added: "We know some team-mates are still to arrive. We train to improve every time and try to do things the way the boss wants.
"You have to have the mentality to play whatever role the boss puts you in. At the start it was a bit different for me in terms of movement, but in the end I felt very comfortable."
On the club's readiness for the season, he said: "The boss has always told it like it is when he's spoken. We know we're strong when we're all united, with the mentality to do great things like we did last year.
"That's something the sporting director and the club have to work out, seeing which pieces are missing for the team, and we have to make sure the dressing room is a good one for all the team-mates who arrive. We're working to have a great season, to fight in every competition, and we're waiting for the players to arrive after the World Cup."
On his new team-mates, Hermoso added: "I've seen Castro and Koulierakis well. They're two young lads who want to do things properly. They're joining a team that, as I said, had a great season last year. They're important pieces to have a great season both in the league and in the Champions League. Every team-mate who arrives to fight alongside us is welcome."
Off the pitch, Roma's pursuit of Feyenoord full-back Givairo Read has stalled, with the Dutch club unwilling to lower their asking price despite Gasperini having already spoken to the player and convinced him to join.
AS Roma have turned their attention instead to Atletico Madrid's Nahuel Molina, whose first offer of around €17m including bonuses was rejected by the Spanish club, with the Argentine himself said to want wages matching what he currently earns in Spain before agreeing to the move. According to Sky Sport, Roma are determined to get the deal done and will improve their proposal to Atletico in the coming days.