The 31-year-old and Brazilian partner Bruno Soares, 35, beat Rohan Bopanna of India and Pablo Cuevas of Uruguay 7-6 6-2 on Sunday.
Murray and Soares will face the USA's Donald Young and Mexico's Santiago Gonzalez for a place in the semi-finals.
In 2016, Murray and Soares won the Australian and US Open doubles titles.
At fifth, the pair are now the highest seeds left in the men's doubles competition at Roland Garros.
"It's a big opportunity for us now to progress far in the tournament," Murray said.
"We'll just do our best to focus on each match and fight for every point like we did today.
"It was a real battle out there, especially the first set was really close, a really high level from everyone I thought. We're really proud of ourselves for what we managed to do today."
Scotland's Jamie Murray has reached the quarter-finals of the French Open doubles for the first time.