The South Americans dominated proceedings at Arrowhead Stadium, registering eight shots on target, but their wastefulness in front of goal meant Arias' 14th-minute finish was all they had to show for an otherwise commanding display.
The breakthrough came after substitute Luis Suárez, who had replaced the injured Jhon Córdoba early in the match, battled down the right flank before delivering a cross that Arias calmly converted.
Ghana threatened in the opening minute through Thomas Partey, whose effort drifted narrowly wide, but the Black Stars struggled to create meaningful chances thereafter. They failed to register a single shot on target throughout the contest and managed less than half of Colombia's completed passes in the first half.
Both teams were forced into early substitutions after injuries to Córdoba and Ghana defender Marvin Senaya, who was replaced by Alidu Seidu.
Colombia continued to control possession and carve out opportunities, with Luis Díaz missing several chances to extend the lead. The Bayern Munich winger fired wide before halftime, later had a goal ruled out for offside, and saw another effort saved by Lawrence Ati Zigi, who also produced an excellent stop to deny Johan Mojica.
Juan Quintero also came close late on, but Colombia's inability to convert further chances was not punished as Ghana rarely threatened.
Named Man of the Match, Díaz admitted Colombia still have room for improvement in front of goal.
> "We're just missing that little detail of finishing a bit better and staying composed," he said. "I think we played a very solid game."
The winger also reflected on his own missed opportunities.
> "I had chances to convert. That confidence I want to have isn't quite at 100 per cent yet. I'm trying to improve game by game. I felt much better in today's match than in the previous one."
Colombia will now face Switzerland in Vancouver on Tuesday for a place in the World Cup quarter-finals, while Ghana bow out after an underwhelming attacking display in the knockout stage.
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