Fredy Guarin scored two minutes from time against a tiring Hoops side, who played for almost an hour with 10 men after Virgil van Dijk was dismissed in the 36th minute.
The visitors had the first opportunity of the match on seven minutes when Gary Mackay-Steven took an angled pass from Emilio Izaguirre and stepped past his marker only to shoot against the legs of the advancing Juan Pablo Carrizo.
Inter responded with a powerful shot from Guarin which flew a couple of foot past Craig Gordon’s right post.
Celtic had a strong penalty claim waved away by referee Ivan Kruzliak when Davide Santon appeared to bundle Stuart Armstrong over as the midfielder got goal side in the box.
Santon then zipped a low delivery just across the penalty area at the other end of the park, but the ball evaded Rodrigo Palacio and Mauro Icardi inside the six-yard box.
Virgil van Dijk had been booked for a challenge on Palacio, when the Celtic defender appeared to win the ball cleanly, and when the Dutchman tangled with Icardi nine minutes before the break he was dismissed on a second yellow.
Inter moved to take advantage with a spell of pressure and came close just before as the break when Jason Denayer threw himself in front of a Xherdan Shaqiri shot and Gordon blocked the rebound from Danilo D’Ambrosio.
The home side continued to press after the break and Palacio crashed over the target from 12 yards under pressure from Efe Ambrose.
Gordon had to scramble to his left to push away a low curling low drive from Hernanes, and the Celtic keeper then thwarted Icardi from close range as he latched on to a reverse pass from Guarin.
And Guarin could have been ordered off when he clashed with Stefan Johansen as Celtic sprung a late counter-attack, but he was given a yellow instead.
Kris Commons almost made an impact two minutes after coming off the bench when he forced Carrizo to parry his dipping shot from 40 yards.
But Celtic’s resolve was broken two minutes from time when Inter struck on the counter-attack and Guarin crashed a brilliant shot from 30 yards past the diving Gordon.