While Barcelona and Sporting Lisbon became the first teams to secure a place in the next round, Liverpool needed a penalty by Steven Gerrard in the fifth minute of injury time to draw 1-1 with visiting Atletico Madrid.
Inter, which hired coach Jose Mourinho in the offseason with the intention of winning the tournament, avoided a particularly humbling defeat when Julio Cruz scored in the 80th minute to salvage a 3-3 draw at Anorthosis Famagusta.
With Panathinaikos winning 3-0 at Werder Bremen, Inter still tops its qualifying group and should still qualify for the next round but the result could raise further doubts about Mourinho's future, especially with the Nerazurri in the doldrums in the Italian league.
If Inter does go on to have a successful season, Mourinho may look back upon Cruz's 12-meter (yard) header from Mario Balotelli's free kick as one of the turning points.
Inter twice led in the first half in Cyprus but Cedric Bardon and Giorgos Panagi scored to equalize goals by Balotelli and Marco Materazzi.
Five minutes into the second half, Inter defender Nicolas Burdisso tried to head away Bardon's right-wing cross but only glanced it into the path of Nikos Frousos, who hit a low shot past goalkeeper Julio Cesar and into the net.
Mourinho, who before this match had watched his team play out three scoreless draws and a labored 3-2 win over last-place Serie A side Reggina, responded by withdrawing Burdisso in a triple substitution.
But Inter needed goalkeeper Julio Cesar to make good saves from Vincent Laban and Andreas Constantinou before Cruz rescued his team.
Inter has eight points in Group B, three more than its opponent. With two games to play, Panathinaikos is a point further back after goals in Germany by Evangelos Mantzios, Giorgos Karagounis and Alexandros Tziolis reignited its slim hopes of advancing.
Chelsea's lead in Group A was whittled down to just one point by defeat to Roma, which led in the 34th through a goal by former Chelsea defender Christian Panucci.
"First, we did not play very well," Chelsea manager Luiz Felipe Scolari said. "Second, we made mistakes at critical times. (We gave) Roma their two best chances and Roma took it and killed the game.
"Roma maybe did not play one or two or three games very well here in Italy, but they have very good players and when you give them chances like tonight they're tough to stop."
Panucci side-footed a cross from Cincinho past goalkeeper Petr Cech and Mirko Vucinic made it 2-0 three minutes after halftime with a shot from the edge of the area through a crowd of players.
The Montenegro striker hit a third goal for Roma 10 minutes later before Chelsea captain John Terry, who missed what would have been the winning penalty kick in last season's final defeat to Manchester United, got a goal back with a close-range shot from a rebound.
Roma and Bordeaux, which won 2-1 at Romanian club FCR Cluj, each have six points to Chelsea's seven.
Atletico and Liverpool stayed tied atop Group D after Gerrard converted a penalty he was awarded when he fell to the ground while challenging Mariano Pernia for a header. The goal cost Atletico early progress to the next round.
"One moment we were in the next phase qualified because of Marseille's win," coach Javier Aguirre said through a translator. "We've got to carry on fighting."
Atletico had led since the 37th, when Argentina midfielder Maxi Rodriguez collected a pass by Antonio Lopez, took one touch to burst past defender Jamie Carragher in the area and shot across goalkeeper Pepe Reina just inside the far post.
Five-time European champion Liverpool, which was again without star striker Fernando Torres because of a hamstring problem, is tied with Atletico on eight points. Marseille beat PSV Eindhoven 3-0 to join the Dutch team on three.
Barcelona advanced from Group C despite being held to a 1-1 draw by visiting FC Basel. Barcelona has 10 points, one more than Sporting - which beat Shakhtar Donetsk 1-0 and is a point further back.
Shakhtar can catch Sporting on points but has lost twice to the Portuguese team and therefore cannot finish ahead.
Defending champion Manchester United is at Celtic in one of Wednesday's eight matches, while four-time winner Bayern Munich is a Fiorentina, Juventus is at Real Madrid, and Arsenal hosts Fenerbahce.
