But Inter retaliated to get one point thanks to the 87-minute equalizing goal of Mauro Icardi, keeping three former winners on course to join Barcelona in qualifying for the last 16 of Group B.
Tottenham remained in the hunt, however, after Kane's late goal secured a 2-1 win over PSV Eindhoven.
The Dutch side started with a superb start at Wembley, taking the lead when Luuk de Jong headed home in the second minute.
Mauricio Pochettino's team began to despair when captain Kane equalized with 12 minutes remaining, and he headed home in the 89th minute with the help of two deflections when Spurs got their first win in the campaign.
"We have to dig deeper and find another level when it's important. Our whole season has been crushing the results here and there," Kane said.
Liverpool were a strong favorite for Red Star in Serbia, but Jurgen Klopp's team was poor in a 2-0 defeat to 1991 European Cup champions.
Milan Pavkov scored twice in the first seven minutes of the first half, headed for the opener and then beat Alisson with a powerful strike from distance, to please the support of the big house.
"Children are very disappointed, I am very disappointed and we have to do better," Klopp said.
The defeat exposed the Anfield club to the threat of early elimination in Group C, with Napoli coming from behind to draw 1-1 at home against Paris Saint-Germain in another match that night.
Napoli hold PSG
Kylian Mbappe arranged for Juan Bernat to take Paris ahead before halftime, but Lorenzo Insigne beat Gianluigi Buffon from the penalty spot in the second half when the two teams drew for the second time in two weeks.
"This is a difficult group but we saw with Liverpool losing that everything was possible," said Julian Draxler of PSG. "The draw at Napoli is not bad and we still have our fate in our own hands."
Liverpool and Napoli are locked on six points from four matches, with PSG one behind and another Red Star adrift of points. Liverpool travel to the French capital in the next match.
Meanwhile, Atletico Madrid were avenged for a 4-0 defeat to Borussia Dortmund last month when Diego Simeone's side beat Germany 2-0 in Spain, Saul Niguez and Antoine Griezmann scored.
Two favorites in Group A were not guaranteed a path to the last 16, however, after Club Brugge claimed a stunning 4-0 win at Monaco.
Belgium haven't won a Champions League match since 2005, but they were 3-0 up midway through the first round at the Stade Louis II. Hans Vanaken scored twice, including a penalty, and Wesley belted Brazil in third before Ruud Vormer sealed the victory.
& # 39; Worry & # 39; for Henry
Semi-finalists in 2017, Monaco are now eliminated. It was a poor performance from a team that had never won in 15 matches, including five matches under Henry.
This result also came on the day Russian billionaire owner Dmitry Rybolovlev was arrested in a corruption investigation.
"We started well, but once we were left behind, we stopped playing and it was a little worrying," Henry said of his team's appearance.
Porto are on the verge of qualifying from Group D after a 4-1 win at home to Lokomotiv Moscow. They led the section with two points from Schalke, who beat Galatasaray 2-0 in Gelsenkirchen.