The result came as the club pressed on with strengthening the squad by agreeing a deal for Real Oviedo left-back Rahim Alhassane.
Bologna trailed 3-0 inside 29 minutes courtesy of a Joel Grodowski brace and a goal from Marius Wörl, before the game descended into chaos on the stroke of half-time.
Rowe, who enjoyed an impressive first season in Serie A after joining from Marseille, reacted furiously to a foul, chasing down his opponent and striking him in the chest before appearing to headbutt him, sending the German player to the ground.
The referee had little choice but to issue a straight red card, and Rowe continued to remain close to the confrontation as tempers flared further, with Santiago Castro and other team-mates stepping in to pull him away and calm the situation.
Maximilian Bauer added a fourth goal for Arminia in stoppage time to complete the rout, with Bologna reduced to 10 men for the last hour of the friendly, played in front of around 2,000 fans at the club's Val Pusteria training camp.
The result offered little encouragement for Tedesco as he begins life as Bologna's head coach, with the Italian-German using a heavily rotated squad across both halves in the club's first pre-season fixture. Bologna will look to put the display behind them when they face fellow German second-division side Heidenheim in their next friendly.
Away from the pitch, Bologna confirmed they have reached an agreement with Real Oviedo for the permanent transfer of left-back Rahim Alhassane, in a deal worth €3.5m.
The move represents a change from the loan structure initially discussed between the two clubs, with Bologna instead opting to sign the 24-year-old outright as a new option down the left flank. According to transfer expert Fabrizio Romano, Alhassane is set to undergo a medical in the coming days before completing his move and putting pen to paper on his contract.
Alhassane, who holds both Nigerian and Niger nationality and represents the Niger national team, joined Oviedo in 2024 and played a role in the Spanish club's return to La Liga after 24 years away from the top flight last season.
He is expected to provide competition and cover for Juan Miranda at Bologna, giving Tedesco a further defensive option as he continues to shape his squad ahead of the new Serie A campaign.