Cruzeiro bucked the trend as they left Argentina with a win and a priceless away goal on Thursday, while Guarani won at home.
Marquinhos was on target as Cruzeiro became the first away team to score during the quarter-finals of the Copa Libertadores, after trumping River Plate 1-0 in Buenos Aires.
No away team in the other three last-eight fixtures had scored on the road but Cruzeiro put an end to that thanks to midfielder Marquinhos and his 82nd-minute strike at El Monumental in the Argentine capital on Thursday.
River - taking part in the quarters after rivals Boca Juniors were expelled from the CONMEBOL competition - came unstuck eight minutes from time, having failed to clear their lines.
Giorgian De Arrascaeta's effort was blocked by River goalkeeper Marcelo Barovero, but Marquinhos was on hand to poke the ball home from inside the six-yard box, much to the delight of the visiting Brazilian fans.
The return leg will take place in Belo Horizonte on May 27.
A solitary goal decided Guarani's hosting of Racing Club, as the former recorded a 1-0 victory in the earlier kick-off.
With the game petering out towards a draw in the Paraguayan city of Asuncion, Julian Benitez unleashed a powerful, long-range shot past Racing keeper Sebastian Saja in the 85th minute.
Benitez - on the run - fended off his opponent as he fired the ball low across Saja from outside the penalty area.
Guarani head to Argentina for the second leg in Buenos Aires on May 28.