Goals from Denis Cheryshev and Ruben Sobrino in Glasgow had left Celtic with a mountain to climb at Mestalla and Toljan's two bookings made their challenge even greater.
The Scottish champions worked hard but with little success and Valencia - with their two-goal and one-man advantage - made sure of their progress through substitute Kevin Gameiro in the 70th minute.
Celtic defender Kristoffer Ajer had two bites at the cherry from a corner in the 10th minute, seeing one attempt blocked before his second was blasted over.
Valencia lost Ezequiel Garay to a knee injury midway through the half but their task became easier when Toljan, on loan from Borussia Dortmund, saw red after taking down Goncalo Guedes as he galloped goalwards.
Scott Bain proved his worth before the interval, with two impressive saves from Dani Parejo and Santi Mina, and Jozo Simunovic blocked a Sobrino strike in the goalmouth six minutes into the second half.
But Celtic's fate would soon be sealed.
Gameiro - two minutes after coming off the bench - got on the end of a Daniel Wass knockdown to prod the ball home from close range with his knee.
The result maintains Celtic's unwanted record of having failed to score in an away knockout match in Europe since April 2003, drawing a blank in nine matches in that period.