The only known instances of helicopters shooting down other helicopters came in the Iran-Iraq war. Iranian AH-1 Cobras engaged and shot down Iraqi Mi-25s on I believe at least two occasions. I do know for a fact that in at least one case, which would be the first ever case if there were indeed more, that the Cobras jumped the Iraqis and downed them with TOW missiles. TOW missiles are generally used as surface to surface or air to surface missiles, but since they are wire guided, this can be adopted for such purposes as downing another helicopter. The first Cobra scored an immediate kill, while the second Cobra's target sustained heavy damage and crashed later upon its attempt to return to base.

There are plenty of reports of further helo vs helo action in the war, but accurate and verified documentation is hard to come by from both sides. That said, it is very likely and probably certain even that there were more engagements with victories being scored, but I do not really know the details.