The billboard's claim particularly disgusted the Condin family, whose 3-year-old son is being treated for leukaemia.
As part of her submission the the ASA, Jody Condin described the sign as "offensive and upsetting''.
"(It's) totally disrespectful and hurtful to those that have lost loved ones to cancer or are caring for loved ones with cancer and waiting for them to die.
"How dare they make a claim like this. It is false advertising.''
She said the message on the sign was conveyed more as fact than a message of belief and hope.
"This is dangerous and deceptive as it could potentially offer false hope and lure in the vulnerable in their time of illness and sadness.''
After going public with her opinion in March, Ms Condin received emails and phone calls from supporters across the country, some of whom had lost loved ones to cancer.
In its written response to the complaints, Equippers Church said it never intended to cause offence with the billboard, but acknowledged it may have done so to some members of the community and as a result it had since replaced the "offending'' statement.
The replacement billboard displayed a summarised sentence of a Bible verse: "Jesus heals every sickness and every disease - Matthew 4:23''.
However, in its response to the ASA, the church said the original claim, "Jesus Heals Cancer'' was an expression of Christian beliefs and was supported by results.
"Our belief is substantiated by the fact six people within our congregation have testified to Jesus healing them from cancer.
"It is also our passion to provide a place of support, hope and healing as we seek to build and encourage people in their relationship with Jesus.''