The Argentina-born forward scored twice in four second-half minutes to overhaul Stanislav Manolev's stunning opener, before Georgi Milanov brought the hosts back on terms in Sofia.
Osvaldo's goals were not up to the spectacular standard he would have liked and the result remained a slight disappointment for the Azzurri, but the Roma player was happy to help his side earn their first Group B points.
"I am happy with the two goals on a memorable night, even if the result wasn't pleasing," he told reporters.
"Clearly it would have been better to win, so I'm half happy.
"It was a difficult game and at times in the first half we did well, while we were under more pressure after the break. Bulgaria played well and the draw was the right result.
"I usually score beautiful goals and these were somewhat scruffy, but they are worth just as much.
"I want to dedicate these goals to my daughter Victoria, who every time she goes to church says a prayer for me and lights a candle, so they are for her."
There were promising signs from Osvaldo's new-look link-up with strike partner Sebastian Giovinco, as well as with Roma team-mate Mattia Destro after he replaced the Juventus man.
"I had some good exchanges with Seba, though at times couldn't find the space. I get on very well with him as a strike partner back from our times for the under-21 side," he said.
"I linked up well with Mattia Destro too. They have different characteristics, as Mattia is more of a centre-forward like me, but he has good feet too for passes.
"At times we were a little slow and did not play the way Italy usually do. Bulgaria put pressure on us and that caused us problems, so at times we managed to pass the ball around and at others we didn't.
"Bulgaria were sitting deeper at one point, then suddenly poured on pressure and made it difficult for us."
Italy coach Cesare Prandelli was happy with the result after admitting his side did not perform to the best of their capabilities against an impressive Bulgaria team.
"There was a lack of pace and accuracy but the point is good because we usually start slowly," he told television station Rai.
"We missed two chances to make it 3-1 but they were good, they upped the pace and put us in trouble."