I have a feeling that what will be involved here is a differing use of the word "cause."
What they may say is that they "caused" Jesus to save them in the sense that by confessing faith in Christ, calling on the name of the Lord, etc., they met the Lord's "requirement" for salvation, and as such He gave them what He said he would give to those who met his "requirement" of faith in Christ.
I don't know, just guessing.