Pochettino was only four months into his first season as Spurs manager and his side, who had lost at home to Newcastle the previous week, were 1-0 down at Aston Villa.
Pressure had begun to mount on Pochettino before a dramatic turnaround at Villa Park as Nacer Chadli equalised with six minutes left and Kane added a final flourish.
The striker, brought on as a substitute, scored a deflected free-kick in added time to give Tottenham, and his coach, a pivotal victory. Pochettino sees it as a turning point.
“Maybe I can find many other goals that were fantastic but for me the goals are related to emotion, in the period you are living,” Pochettino said.
“For me that goal was an amazing goal because it meant for us, for everyone, the possibility to stay here today.
“It’s true, the team were 14th or 13th and we were a little bit under pressure because the team were there and play well, but not win games.
“Always in football, when you don’t get good results, the first to be sacked is the manager. That is why that goal allowed us to carry on working and try to change the club.”
Kane will be looking to continue his rich vein of form against Swansea on Saturday after his double against Borussia Dortmund in midweek took his tally to six goals in four matches for club and country.
Port Vale boss Neil Aspin is expected to choose from an unchanged squad for the home game against Exeter.