The 31-year-old, ranked a career-high 25 in the world, battled past Frenchman Mannarino 6-4 7-6 (7).
In doing so Evans joined Cameron Norrie and Jack Draper in quarter-finals, meaning three British players have reached the last eight at Queen's for the first time in the Open era.
Evans said: "It felt like a high-level match. I'm just pleased to have come through.
"He's good on grass, I knew it was going to be a difficult match but I didn't envisage it being that tight and I obviously didn't want it that tight."
Evans looked in for an easy afternoon when he took the first eight points of the match but Mannarino, the world number 42, proved a tricky customer once he settled.
Evans needed a further break to take the first set and saved a set point in the second before winning a closely-fought tie-break.
Evans added: "It was a tough match mentally more than anything, to try and hold on and hold my nerve at the end as well, I was down set point.
"I didn't seize the moment very well at 3-2, I don't think I got off the ground. It's difficult to play your game, go after it, you want to be safe sometimes and make him miss.
"I thought I did a good job of playing my game in the end and came up with some good serves when I needed them."
Evans' victory also meant three Britons have reached the quarter-finals of an ATP Tour event for the first time since Nottingham in 1996, when Tim Henman, Greg Rusedski and Mark Petchey got through.