Antonio Conte hailed Inter's "spirit" as they kept alive their chances of progressing in the Champions League with a pulsating 3-2 win away at Borussia Monchengladbach.
Inter put aside the disappointment of failing to win any of their first four Group B matches to edge past Gladbach, who could have sealed their own qualification for the knockout stages with a win on home turf.
Matteo Darmian put the Serie A side ahead with his first goal for the club before Alassane Plea pulled the hosts level on the stroke of half-time.
Lukaku then scored two fine goals to put Inter 3-1 up and despite Plea's second 15 minutes from time, Inter held on to ensure they go into the final round of fixtures next week with something to play for.
They will need to beat Shakhtar Donetsk and hope Gladbach do them a favour against Real Madrid, who lost for a second time this season to the Ukrainian side in the early kick-off on Tuesday.
Inter were good value for the win too, carving out 15 shots on goal to Gladbach's 10, and Conte revealed a supportive team talk at half-time was crucial to their success.
"Gladbach are in an excellent period of form, in both the Bundesliga and the Champions League," Conte told Sky Sport Italia . "If you are not a team, tactically, physically, in terms of character, you don't come here and win.
"We absolutely deserved to win and could've frankly avoided suffering quite so much towards the end. We beat the group leaders and we are alive. I thank the lads; they had the right spirit. When they have this desire to help each other out and be a real team, we cause problems for the opposition.
"What I ask of the lads is to increase their level of responsibility, each and every one, because if they all do that, then we can become an ambitious side.
"Performances like this prove the team is there, we are united, we're all working in the same direction and want to give our fans joy.
"Conceding that equaliser on the stroke of half-time against the run of play could've killed us. We analysed some situations during the break, as we made some mistakes towards half-time, but these lads needed my support at that moment.
"I told them we had to keep going and rediscover the quality we had early in the game and that we'd be rewarded with the victory. I am glad that I was proved right, because when a coach talks and those things don't happen, you risk losing their trust!"
Lukaku's double means he has now scored 12 goals in European competitions over the past two seasons, a tally only bettered by Robert Lewandowski and Erling Haaland.
The Belgian, who scored with his only attempts on target, joined Inter in a big-money move from Manchester United in August 2019 and says his time in Italy has been the most enjoyable of his career so far.
"I am one of the 25 players," he said. "There are far more experienced players here than me. I want to help the team to win, that's what I do in every training session and every match. I want to help Inter. I am so happy to be here with these lads. I've had the best year-and-a-half of my career at Inter, so I thank the coach, his staff and my team-mates."
Inter take on Bologna in Serie A this weekend before their crunch clash with Shakhtar in Europe.