Bouchard made it to the last four of the Australian Open earlier this year, and her impressive form in 2014 continued at Roland Garros on Tuesday.
The 18th seed prevailed in a topsy-turvy encounter, with both players gaining the initiative at various points in the match.
Suarez Navarro looked primed to take the first set before crumbling when on the brink of taking the lead and, although she hit back in the second, a late break from Bouchard in the decider set up her victory.
Bouchard will now take on seventh seed Maria Sharapova in the semi-final, aiming to shock the Russian and reach her first grand slam final.
Suarez Navarro drew first blood on the Paris clay, and was serving for the first set at 5-2 when Bouchard found the net with two backhands, however, the Spaniard seemed beset by nerves as a result and allowed her opponent to hit back.
The 14th seed seemed unable to shake her apprehension, wasting a set-point opportunity as Bouchard claimed four games on the bounce.
Suarez Navarro managed to stop the rot by holding and for the set to a tie-break, only for Bouchard to break four times and come out on top.
The pair exchanged breaks early in the second set, but Suarez Navarro took the initiative in game six, setting up break point with a superb backhand winner.
Buoyed by that, the Spaniard took each of the next two games and took advantage of her second set point to send the match into a decider.
Once again Suarez Navarro moved into a commanding position, but as in the first set she fell apart, and allowed Bouchard to level having been 4-1 behind.
The Canadian reeled off 12 consecutive points, taking three games to love, setting up a match-winning opportunity by breaking her opponent.
After wasting two match points, there was no mistake from Bouchard on the third as a deep forehand forced Suarez Navarro to return into the net.