Kucka's header in the 21st minute bounced off the post and hit goalkeeper Christian Gratzei's back before it went over the line. It was Kucka's first goal in 17 matches for Slovakia.
The unmarked Jez made it 2-0 nine minutes later though TV replays showed he was offside when picking up a pass by Filip Holosko.
Erwin Hoffer reduced Austria's deficit with a header from Florian Klein's cross in the 62nd, seven minutes after coming on as a substitute.
Hoffer saw a second goal disallowed for offside in the closing minutes.