The Blues did secure a potentially vital away goal through Salomon Kalou in the early stages of the second half, but they were unable to claw their way back onto level terms for a second time in a game where they had plenty of possession but created few clear-cut chances.
Inter coach Mourinho had stoked up the flames ahead of the meeting with his former club but it was Carlo Ancelotti who raised eyebrows with his team selection.
The Italian tactician opted for a three-pronged attack of Didier Drogba, Nicolas Anelka and Kalou, while Florent Malouda started at left-back in the absence of both Ashley Cole and deputy Yuri Zhirkov.
However, it was Mourinho and Inter who were celebrating in the early stages of a cagey clash at the San Siro as the Italian champions raced into a third-minute lead.
Thiago Motta's throughball found Diego Milito inside the Chelsea penalty area and the Argentina striker took full advantage of John Terry's hesitancy.
Milito pulled the trigger from 12 yards and found the back of the net with a thunderous finish.
Drogba rattled the crossbar with a blistering 25-yard free-kick 10 minutes later as Chelsea began to find their attacking feet.
The visitors should have been 2-0 down on 33 minutes when the normally clinical Samuel Eto'o thrashed at fresh air from six yards with the goal gaping.
Inter too survived a scare when the referee inexplicably ruled that Walter Samuel had not clipped the heels of Kalou when he raced in on goal on the stroke of half-time and waved away angry penalty appeals as the players headed down the tunnel at the interval.
Chelsea were back on level terms six minutes into the second half when the largely ineffective Kalou profited from a goalkeeping mistake.
Julio Cesar made a hash of turning away his curling shot from the edge of the penalty area and the ball nestled in the back of the Inter net to silence a raucous San Siro crowd.
However, Inter were on back in the ascendancy just four minutes later as Chelsea failed to deal with the powerful left-foot of Esteban Cambiasso.
The Argentina midfielder was given two bites of the cherry and his low angled strike powered into the bottom corner of Petr Cech's net after the first shot had deflected back off Terry.
Just as worrying for Chelsea was the sight of Cech being stretched off in the 62nd minute with suspected knee ligament damage when he appeared to catch his studs in the turf.
Henrique Hilario came on to replace the Czech Republic international but he had little to do as Chelsea finished the stronger of the two sides.
A subdued Frank Lampard wasted his side's best chance to equalise when he scuffed a low shot at the end of a superb team move and Julio Cesar was able to scramble his close-range shot away to safety.
Chelsea must now hope that they can raise their attacking game in the second leg at Stamford Bridge in three weeks' time if they are to progress at the expense of Inter and their former manager Mourinho.