The church was St. Joseph parish (a Roman Catholic church) on South Saginaw Avenue in Chicago. I found the record on the Family Search site. I searched a database called:
Illinois, Chicago, Catholic Church Records, 1833-1925
Anna's birth record was found in the database called "Baptisms 1913-1925", image 66 of 169.
The marriage record of her parents was found in the "Marriages 1900-1915" database, image 51 of 59.http://boards.ancestry.com/localities.scan-balt.lithuania.ge...
I don't know if the entire congregation was Lithuanian. I suspect it might have been a mix of nationalities. This branch of my cousin's family is definitely Lithuanian, although some censuses call them Russian.