Security alarms tend to "go off." Bells tend to sound. And alarm clocks tend to ring. So while the meaning of your alterations would not be misconstrued, it would certainly sound comical, especially with "rang." "Went off" is the most appropriate phrase here.
Flash is correct. However, a writer may substitute "alarms began to ring" or "alarms sounded," depending on his personal taste or on the feeling that he wants to convey to the reader. Eg., Suppose that the script revealed that the alarms had not gone off during tests. Adjustments were made. Then the burglar set them off. The "alarms sounded" almost states a relief. Then, ring or ringing? From the burglar's point of view, ringing would be a nagging sound and "alarms began to ring" would bear out what the burglar must be going through.