In the Portuguese's first home game in charge, Spurs appeared on course for a disastrous night in Europe having fallen 2-0 behind against their Greek opponent after just 19 minutes.
The north London outfit were not at the races for much of the first-half but a defensive error by Olympiakos just before half-time gifted Dele Alli a goal and handed Tottenham a lifeline.
Spurs scored three times after the interval, with Harry Kane netting twice as he became the fastest player to reach 20 Champions League goals, needing just 24 games to hit the mark and break the record of Juventus legend Alessandro del Piero.
Serge Aurier was also on target in the 4-2 triumph which guaranteed Spurs' place in the last-16 of the Champions League with a game to spare.
Having been staring at the possibility of needing something from their trip to Bayern Munich in the final round of group matches in a fortnight when trailing, Mourinho was delighted to finish the job of qualification at the earliest opportunity.
The ex-Manchester United and Chelsea boss says he did not read the riot act at the interval after a poor first 45 minutes and instead told his players to regain their composure.
"I did not expert our boys to feel this pressure at home and they have not had the best results in the last few months," said Mourinho.
"It was a tough time, more than the changes or the tactical talk, I told them to keep calm, keep confident and believe in ourselves."
Mourinho went on to apologise to Eric Dier having made the decision to take the England international off after less than half an hour of Tuesday's fixture.
Christian Eriksen was introduced in his place and played a major part in the side's turnaround, and Mourinho says Dier "understands" why he needed to be sacrificed for the good of the team.