Ivorian midfielder Kouko Guehi powered in a header from a corner in the 95th minute to earn the Moroccans victory after the game finished 1-1 the end of 90 minutes.
Raja will now play South American champions Atletico Mineiro, featuring former AC Milan and Barcelona forward Ronaldinho, in the last four in Marrakech on Wednesday.
Mineiro will hardly relish the prospect of facing Raja and their boisterous fans who created a pulsating and intimidating atmosphere for the whole two hours at the Agadir stadium.
Monterrey, taking part in the Club World Cup for the third time in a row, were left to contemplate another in the line of Mexican failures at the tournament.
Mexican sides have played at every edition since 2006 but, despite coming from one of the most lucrative leagues outside Europe, none have reached the final.
Raja made a confident start and Zakari El Hachami prodded wide after an incisive pass by Mouhssine Iajour.
They went ahead in the 24th minute when Monterrey goalkeeper Jonathan Orozco fumbled El Hachimi's cross from the right and Chermseddine Chtibi slipped home the rebound
The Mexicans levelled in the 53rd minute with a simple goal when Humberto Suazo floated over a free kick from the right and captain Jose Maria Basanta headed in at the far post.
Monterrey should have gone in front when Cesar Delgado lofted a pass over the Raja defence to send Neri Cardozo clear on goal, but his shot was saved by Askri.
Raja came back into the game and got the winner five minutes into extra time when Guehi rose above the Monterrey defence at a corner and powered a header into the roof of the net.