The in-demand Paul Pogba powered home a header in just the fourth minute to suggest the Turin giants were on for a big win.
That failed to materialise, though, as a new Juve strikeforce of Simone Zaza and Paulo Dybala struggled before the hosts grabbed a deserved equaliser late on through Adam Buksa.
But Mandzukic - who joined from Atletico Madrid last month - had the final say, lashing home in the final minute.