Murray and Evans defeated Belgium's Sander Gille and Joran Vliegen 6-3, 6-7 (8/10), 11-9 to qualify for the quarter-finals at Roland Garros.
They saved two match points in the final-set tie-break, two days after they had saved five match points in their opening-round victory.
In a rollercoaster clash on Court Suzanne Lenglen, the British pair wasted two match points of their own in a gripping second-set tie-break.
The 37-year-old Murray, who has claimed three Grand Slam titles, reached the world number one spot and pocketed two Olympic singles gold medals, will retire once his Paris campaign is over.
