Five-times South American champions Penarol of Uruguay joined them in second place in the group despite a 1-0 home defeat by Argentina's Independiente marred by crowd violence at the Centenario.
LDU's Argentine striker Barcos, who had laid on the Ecuadorian team's opening goal for Luis Bolanos two minutes after the interval, embarked on a run down the left flank in the 58th.
Skipping past two defenders near the left corner flag, Barcos ran into the box close to the goal-line and slotted in a shot between goalkeeper Sebastian Torrico and the near post.
"I'm really happy with the move and the finish which came off so well," man-of-the-match Barcos told Fox Sports.
Competition debutants Godoy Cruz, who had a good first half but felt the effects of the high altitude in Quito as the match wore on, had winger Carlos Sanchez sent off in the 67th minute for kicking an opponent on the ground.
At the Centenario, Penarol fans unfurled a 15,000 square metres flag that covered nearly two whole stands celebrating their team's 120th anniversary, but record seven-times champions Independiente spoiled the party.
Striker Facundo Parra scored the only goal after 33 minutes for a second string Independiente side that travelled to Montevideo needing a 8-0 victory to qualify.
Parra played the last six minutes in goal after goalkeeper Fabian Assmann and Penarol striker Juan Manuel Olivera were sent off for fighting.
There was fighting and stone throwing before kickoff between about 100 travelling Independiente fans and Penarol supporters. Police quickly stopped the fighting and there were no arrests.
In Group Five, Chile's Colo Colo went top with a 2-1 home win over Deportivo Tachira of Venezuela.
Striker Diego Rubio scored both their goals after midfielder Edgar Perez Greco had given Tachira a third-minute lead.
The Chilean team are a point ahead of Paraguay's Cerro Porteno who host third-placed Santos in Asuncion on Thursday in a match the Brazilian side must win to retain a realistic chance of qualifying.
