The Primera Division side was leading 3-1 from the play-off first leg and looked to be cruising through when Alvaro Negredo took advantage of a defensive mistake to put Valencia, runner-up in 2000 and 2001, ahead after just four minutes with a sublime chip.
However Monaco, which lost to Porto in the 2004 final, fought back and Andrea Raggi (17) and Elderson (75) both netted past Australia goalkeeper Ryan.
The goals set up a grandstand finish at the Stade Louis II but Los Che clung on to create UCL history.
Valencia a 4-3 winner on aggregate saw Spain become the first country to have five teams in the group stage in a single season - the others being Barcelona, Real Madrid, Atletico Madrid and UEFA Europa League champion Sevilla.
