The Black Stars failed to match an organised and confident Colombian side that controlled much of the contest to secure their progression.
Colombia suffered an early setback when striker Jhon Cordoba was forced off through injury after just eight minutes, with Luis Suarez replacing him. The change did little to disrupt the South Americans, who continued to dictate possession and create dangerous attacking opportunities.
Ghana showed early intent when captain Thomas Partey fired narrowly wide in the opening minute, offering hope that the Black Stars could trouble Colombia. However, the Africans were also forced into an early substitution as Alidu Seidu replaced the injured Marvin Senaya.
The breakthrough arrived in the 14th minute. Suarez showed determination down the right flank before delivering a pinpoint cross for the unmarked Arias, who calmly guided the ball into the net to hand Colombia a deserved lead.
The South Americans remained in control for the remainder of the first half, with Luis Diaz causing constant problems on the left wing. The Bayern Munich attacker should have doubled the advantage in the 39th minute but dragged his effort wide of goal.
Ghana goalkeeper Lawrence Ati Zigi kept his side in the contest just before the half-time break, producing an excellent save to deny Johan Mojica's powerful downward header.
Colombia wasted several opportunities to seal the result after the break. Diaz had a goal ruled out for offside before firing another effort straight at Zigi, while Juan Quintero also missed the target late on.
Despite pushing for an equaliser in the closing stages, Ghana could not find a breakthrough as their World Cup campaign ended in the Round of 32, while Colombia advanced to the last 16.
Ghana coach Carlos Queiroz was disappointed by the result but admitted the best team probably won on the night.
"They controlled much better the game. The passing, the movements, and doing that, they damaged our fitness and our ability to recover the ball and try to go forward," he said.
"We tried to go forward, tried to create opportunities, but for any reason, the last pass, the last service, was not with the quality that was necessary in this case. So, I think the best team won."