CHEU is the sister union to the California Nurses Association, which engaged in a months-long battle last year with hospital officials over whether it can represent registered nurses.
CHEU officials say they have authorization cards signed by more than half of the hospital's 1,000 ancillary workers.
Hospital administrators must determine which ancillary employees will be covered in the bargaining unit before state mediation services can count the cards, according to CHEU Director Fernando Losada.
In addition to Wednesday's vote, CHEU is also seeking to represent 125 employees at Marian's satellite facilities, including the Extended Care Center, a large long term care facility.
In the Glendale election, CHEU won 244 votes to 217 for "no union" and 50 votes for SEIU.