Hamilton started 10th in Sunday's Italian Grand Prix after a grid penalty and finished sixth, with Ferrari fans showing him plenty of love on his Monza debut in the red overalls worn by so many greats before him.
Vasseur said the support had been a boost and Hamilton's performance in following former Mercedes teammate George Russell - a race winner in Canada - across the line suggested the results would come for the 40-year-old.
"Yes, we can expect him to be on the podium," said the Frenchman, who had also expected Ferrari to be on the podium at Monza.
Ferrari are second in the constructors' standings, a massive 337 points behind runaway leaders McLaren, but the only team in the top four yet to win - other than Hamilton's victory in a Saturday sprint in Shanghai in March.
There is no podium celebration after the sprint.
"I think the energy he received from the tifosi on Wednesday, Thursday in Milano was something very special for him," Vasseur said of Hamilton, who joined from Mercedes in January and had previously raced at Monza as a Ferrari foe.
"It was something mega and I think this gave him an extra boost all over the weekend."
Hamilton's teammate Charles Leclerc was fourth on Sunday, at a circuit where he won last year, but qualified on pole in Hungary and has a season's best of second in his home Monaco Grand Prix.
Vasseur said the gap to McLaren, beaten by Red Bull's Max Verstappen on Sunday, had been close all weekend.
"I think we were in better shape this weekend, not enough because they are in front of us, but I think at least we closed the gap," he added.
"I think we have a couple of tracks where we are supposed to be also in good shape, starting with the next one. We are historically always in good shape in Baku.
"But for sure the goal is to finish P2 in the championship. McLaren is on another planet."
Vasseur said Ferrari were now fully focused on their 2026 engine and car, and expected other teams to have turned off the development tap.
"I think only one team brought an upgrade this weekend except the wing, because we have to adapt the car to the level of downforce, but I think for everybody now we are focused on 2026," he added.