Giovani Lo Celso opened the scoring on 19 minutes with a curled free-kick before Lautaro Martínez doubled the lead from the penalty spot on 31. Jordan pulled one back through Mousa Al-Tamari on 55, but Messi settled it with a trademark free-kick on 80 to seal the group with a perfect record.
Scaloni made nine changes with Argentina already assured of top spot in Group J, giving minutes to players who had been waiting for their chance in the tournament.
Argentina coach Lionel Scaloni was satisfied with how the match unfolded, singling out several players by name.
"I'm happy because we gave everyone minutes, and that's very important for us," Scaloni said.
"Especially for Lo Celso, who couldn't be at the previous World Cup, and for Lautaro, who hadn't scored yet and had a good night. And Leo's goal. I'm happy, and now the good part begins."
Atlético Madrid midfielder Thiago Almada reflected on a group stage in which Argentina showed they remain the team to beat in the competition.
"We got nine points, we finished first like we wanted, and that's not easy at all," Almada said. "Now it's time to rest and start thinking about the next match against Cabo Verde."
Argentina face Cabo Verde in the round of 32 on July 3 in Miami, where Messi will be the story once again. His record in the tournament continues to build with every appearance, and the holders head into the knockout phase with a squad that has been used intelligently across three group games.
Jordan showed genuine quality in patches despite the scoreline. Their second-half goal from Al-Tamari, worked neatly from defence to finish, was a reminder of why they were not simply making up the numbers. The Hashemite Kingdom depart the tournament having made their World Cup debut with competitive performances, even if they could not find the results to advance.
For Argentina, the perfect group stage conceals some complexity ahead. The holders are in a demanding half of the draw, with fewer rest days than some rivals if they go deep in the tournament. Scaloni's management of minutes across the group stage — resting Messi until after half-time, rotating heavily — was a deliberate investment in freshness for the rounds ahead. The message was clear: Argentina are not here to impress, they are here to defend the title.