Air carrier Captain reported the flight crew initiated the gate pushback prior to sending the DG acknowledgement message. During pushback the flight crew acknowledgement message was sent and a Dispatch confirmation message was received shortly thereafter.

Narrative

During the preflight process; the ACARS system worked normally for everything except the final dangerous goods document. We were never able to receive the final DG paperwork over ACARS. I clipped the preliminary dangerous goods report to my yolk; and I asked the Officer to help me remember not to release the brakes before we got the dangerous good document. We ended up having the ground crew print out the final; DG document and pass it up to the First Officer through his cockpit window. After reviewing the document; and being satisfied that his window was closed; and that the before push checklist was complete; I looked over at the First Officer and said call for pushback clearance. After communication with the ground crew; I released the brakes so that they could capture the aircraft however; we had not yet sent the dangerous goods acknowledgment. As I was releasing the brakes the First Officer was sending the dangerous good acknowledgment through ACARS. I had forgotten about sending the acknowledgment; but he remembered. He just couldn't get it sent before I released the brakes. We got the message that said we had released the brakes prior to sending the DG acknowledgment and then shortly there after got the message that dangerous goods had been acknowledged.