Defending champions United extended their unbeaten run in the competition to 20 matches but they still have a lot to do at Old Trafford, while Arsenal will feel they should have killed off Roma after a one-sided contest at Emirates Stadium.
Two other ties remain delicately balanced after Lyon held Barcelona to a 1-1 draw at Stade de Gerland and Atletico Madrid played out a 2-2 draw with Porto in Spain.
Manchester United dominated the first half of their clash with Inter Milan at the San Siro but were unable to make the breakthrough as Cristiano Ronaldo saw a header well saved by Julio Cesar before sending another just wide.
Much had been made of the battle of wits between Jose Mourinho and Sir Alex Ferguson in the pre-match build-up and the Portuguese tactician had instilled a greater urgency into his players as the second period got underway.
Adriano spurned a glorious opportunity and Edwin van der Sar was called into action a number of times, although United kept pressing forward and almost sealed victory in stoppage time when Cesar blocked Ronaldo's ferocious free-kick.
Arsenal were the only winners of the evening as they turned in a fine performance at home to Roma to open up a slender lead ahead of the second leg in the Italian capital.
Arsene Wenger was looking forward to returning to European football after three consecutive goalless draws in the Premier League and his patience was rewarded eight minutes before half-time as Robin van Persie put Arsenal in front from the penalty spot after being brought down himself by Philippe Mexes.
It was a familiar story of frustration thereafter as the Gunners continued to apply pressure but Nicklas Bendtner and Emmanuel Eboue, starting in place of the ineligible Andrey Arshavin, squandered gilt-edged chances.
The Giallorossi created one or two openings but Manuel Almunia was rarely tested and Arsenal will be confident of protecting their advantage at the Stadio Olimpico.
Barcelona had to come from behind to earn a draw against Lyon, who were full of confidence after a fine run in Ligue 1.
Barcelona have been in exhilarating form all season and the stage was set for the Catalan side to underline their credentials as one of the favourites for the competition, but Lyon clearly had not read the script as they took a surprise lead after seven minutes when Juninho's clever free-kick from wide on the left deceived goalkeeper Victor Valdes and flew straight into the top corner.
Lyon continued to impress but could not add a second and they were made to pay as Barcelona, who had gradually been allowed to come back into the game, equalised through a Thierry Henry header on 67 minutes.
Atletico Madrid were twice pegged back by Porto in an entertaining encounter at the Vicente Calderon.
Atletico, unbeaten in the group phase as they qualified behind Liverpool, opened the scoring in only the fourth minute with Maxi Rodriguez latching onto a defence-splitting pass from Sergio Aguero to fire in a shot from close range.
Lisandro Lopez had a goal disallowed moments later but he drew Porto level on 22 minutes as he capitalised on a defensive mistake and his firm drive was too firm for Leo Franco to handle.
Diego Forlan restored Atletico's advantage on the stroke of half-time thanks to a terrible piece of goalkeeping by Helton, who fumbled the Uruguayan's speculative strike from distance into his own net.
Lisandro prodded home Aly Cissokho's excellent left-wing cross in the 72nd minute to make it 2-2 and leave the tie in the balance ahead of the second leg in Portugal next month.