The Nerazzurri cruised to a comfortable 3-1 victory at Partizan Belgrade, courtesy of Rodrigo Palacio's brace and a Fredy Guarin effort, with Nemanja Tomic replying for the hosts with a fine volley at the death.
Triumph in the Serbian capital not only guaranteed progression from Group H but also completed a run of 10 straight wins in all competitions for Stramaccioni's men, who now turn their attentions back to Serie A and Sunday's trip to Atalanta.
He told inter.it: "It's a big achievement because it shows the great mentality and character this team has.
"It's an achievement we're proud of but we're not getting carried away.
"In the first half we were comfortable but we upped the tempo after the break.
"I think Jonathan did a brilliant job today to contain (Lazar) Markovic.
"(Matias) Silvestre was very alert, (Marko) Livaja excellent, (Samir) Handanovic pulled off some incredible saves and Palacio produced a top display.
"We're a bit more relaxed now knowing that we've sealed qualification but we want to win the group and we'll keep playing as Inter can.
"Which win marked the turning point for us? It was a defeat, to Siena: that's where I saw what these players are made of.
"But we're looking forward now and thinking about the Atalanta game. It's going to be a very difficult match."
Veteran full-back Javier Zanetti spoke to Inter Channel after the game, adding: "We came here with the aim of going home with qualification in the bag, and now that we've done it we can relax.
"What's more, we achieved it at a tough ground.
"We managed to adapt to the different phases of the game, which is what the coach asked us to do."
Silvestre also assessed the Nerazzurri's performance, saying: "We had to be careful and remain focused. It wasn't easy but we did it.
"Handanovic was superb. We wanted to go through and now we have."
Inter's Slovenian goalkeeper may have been flattered by his team-mates' compliments but focused on analysing the overall team display.
"It was good to get the win," Handanovic said.
"We couldn't come here and expect them not to have shots on goal but I think it was our experience that won it for us in the end.
"They have lots of young players and they play well, but we made our greater experience count.
"I wouldn't say there was one game in particular.
"All the games we've won have helped us, and the Europa League has given us a hand too because we started earlier so we've hit form quite quickly compared to our rivals."