But three second-half scores from Michele Rizzo in the 42nd minute, Benjamin de Jager five later and Tobias Botes in the 57th gave the Italians momentum.
Three penalties and a conversion from Kris Burton to three and two from Dan Biggar saw the score placed at 26-23 in Treviso's favour going into the dying minutes.
However 19-year-old Morgan, making his Heineken Cup debut, salvaged a draw for the Ospreys with just seconds on the clock slotting a 40-metre penalty.
Director of rugby Johnson's disappointment at the result was clear to see and he believes they may have taken their RaboDirect PRO12 counterparts, who sit sixth in the table, too lightly.
"Maybe we're not giving enough credit to Treviso," said Johnson - whose side remain unbeaten in second in Heineken Cup Pool 5.
"Their form has been pretty good, and it's not all that long since they put 50 points on the Newport Gwent Dragons.
"I'm hopeful that other sides will under-rate their quality. They can still have major say in the fate of this group.
"I was very disappointed with our display, and I was very disappointed with our skill set. It wasn't perfection but our destiny is still in our own hands, and that's not a bad thing."