I can't say I know anything about Cohen's intended meaning,only he could lift the veil on that, but at first blush, (I really love this song), I see many scripture references...

"slave" reference: (this is just one of many referring to being a "slave to the Lord, rather than slave to man...) Rom 1:1 LITV Paul, a slave of Jesus Christ, a called apostle, separated to the gospel of God,

"Roll away the stone"-(again, several scriptures regarding this, this is just one) Matt 28:2 There was a violent earthquake, for an angel of the Lord came down from heaven and, going to the tomb, rolled back the stone and sat on it.

"where the word became man"- John 1:14 The Word became flesh and dwelt among us...

"suffering began"- Jesus' suffering in Gethsemane Luke 22:39-46

"troubles came...I saved"- (this might be a big stretch, but...) Psalm 138:7 speaks of when our troubles came, He reaches out to save us...

My point being: I hear a prayer in song...and perhaps another stretch, Cohen is calling out to be led back to that place of being a slave (to the Lord?) and the Lord answering how He has saved him and loved him as such...