Spezia were left to rue their mistakes after several great scoring opportunities, including a Daniele Verde effort cleared off the line, as Udinese won it with debutant Lazar Samardzic.
The Aquilotti were convinced this would be their first win of the season under coach Thiago Motta, and they certainly had the chances to get the three points.
Viktor Kovalenko, Kevin Agudelo and Verde all threatened, with Verde denied at the 86th minute by a Bram Nuytinck goal-line clearance.
However, against the run of play, 19-year-old Samardzic rode a tackle to give Udinese a last-gasp victory.