The 23-year-old England worldwide midfielder started the match and played for the entire 90 minutes, as Jose Mourinho's side won 4-2 to book their pace in the last-16 stage of Europe's premier club competition.
Mourinho, who was hired to replace the sacked Mauricio Pochettino on November 19, was overseeing his first Champions League match in charge of Spurs and found his team quickly down two goals inside of 20 minutes after Youssef El Arabi and Ruben Semedo scored for the visitors.
Kane's double took him to 20 Champions League goals in 24 games.
With Bayern Munich also sailing to victory, it would have taken a significant result to delay Spurs' passage to the next round.
Mourinho, who won the competition with Porto and Inter Milan, shook hands with each of his players and applauded the fans who applauded back, if not singing his name.
Despite a hard start to the group stage, Spurs still boast two of the top five goalscorers in the competition this season in Harry Kane and Son Heung-min.
The plan paid off as Spurs rallied in the second-half after Dele Alli's brought it back to 2-1 just before the interval.
England midfielder Dele Alli's goal in the dying stages of the first-half gave Tottenham hope, but the game-changing moment came on the 50th minute.
"It was a disappointing start, we know we didn't come out with any energy".
But the challenge he faces at Tottenham immediately became apparent as they began in the sloppy fashion that has been a hallmark of their season so far.
Soon after it was 2-0 to Olympiacos - Semedo tapping in Guilherme's low delivery.
It could not have started worse as the hosts conceded twice inside 19 minutes to Greece's Olympiakos.
He also apologised to Dier for hauling him off early. With the match level at 2-2 and Olympiakos attempting to regroup as the ball drifted out of play, the youngster quickly launched a ball to Serge Aurier, whose throw-in down the wing found Lucas Moura and the Brazilian picked out Kane for the equaliser.
Suddenly Spurs were rampant and they went ahead in the 73rd minute when Aurier connected sweetly to thrash home a right-footed volley from an angle.
Alli then turned provider in setting up the third goal, with his cross converted by Aurier and Kane's header from Eriksen's free kick completed the fightback.