With 21 minutes on the clock Neumann was brought down in the area and Cirio converted the resulting penalty to give the hosts the lead.
Adelaide doubled their advantage four minutes later when Awer Mabil's shot came back out off the post to the unmarked Neumann, who tapped the ball in.
Melbourne improved in the second half and pulled a goal back through Finkler with 11 minutes remaining.
The visitors pushed hard for an equaliser and were rewarded in the 94th minute when Troisi's shot went in off the crossbar.
The draw moves Adelaide a point clear at the top of the A-League table.