Dodo gave Inter a thoroughly deserved half-time lead in the UEFA Europa League Group F fixture, but Saint-Etienne had the better of the second half and seized a point courtesy of Moustapha Bayal Sall's effort following a corner.
The result left Inter four points clear of the other three teams in Group F, with two matchdays remaining.
"I am proud because we played great football in the first half," Mazzarri told Inter's official TV channel. "We could have and should have put the game out of reach.
"We suffered a bit in the second half but we dug in and brought home this important point, which puts us on the brink of qualification.
"It's a pity we drew. If we had been a little more careful we would have had the full haul, but the lads have done very well and are happy with the response after the Parma game (Inter lost 2-0 at the weekend)."
Mazzarri handed Bonazzoli his first Europa League start, and the teenager was a constant torment to Saint-Etienne before the break.
"He really deserves the compliments," said the Inter coach. "He helped the team and played very well. He worked very hard and allowed us to have the success we did in the first half.
"You could see that he has a future, as long as he continues to work and be professional."
Juan Jesus also saluted Bonazzoli and fellow teenager Andrea Palazzi, who came on in the second half.
"Bonazzoli and Palazzi gave a good account of themselves," said the Brazilian. "We must give them peace of mind because they have the quality."
