The second-leg finished 0-0, so allowing Spurs to progress courtesy of their 1-0 first-leg victory.
“The team put in a fine performance tonight and I think over the two legs we deserved to go through,” said Allegri.
“Unfortunately, that mistake in the first leg cost us dearly. This evening my Milan did everything it possibly could, showing intensity and technique. We only lacked something in the finishing.
“This performance must give us more confidence going back into the Serie A race, as we have been improving for a while now.
“I still cannot be satisfied, as there is a great deal of bitterness at going out, but I am comforted by the way my lads fought.”
The best scoring opportunity saw Robinho’s scuffed shot cleared off the line by William Gallas.
“Usually the players don’t make these purely technical errors, but this was a difficult game,” noted Allegri.
“I am happy for Clarence Seedorf, as I really liked him in front of the defence. Zlatan Ibrahimovic lacked something in terms of finishing, but all the front three did well and were only missing a goal. They moved brilliantly.”