Kane came off the bench to score his 50th goal for the club and prevent Spurs being knocked out of the competition by Leicester in successive seasons.
The Foxes came within minutes of claiming another cup victory at White Hart Lane after Marcin Wasilewski and substitute Shinji Okazaki had cancelled out Christian Eriksen’s early opener.
The draw means Pochettino’s side face two more games against Claudio Ranieri’s team before January is out, with a league fixture at home to the Foxes up next on Wednesday before the replay at the King Power Stadium a week later.
“It’s complicated because it’s tough to play in the Premier League, the Europa League, and the cups,” said Pochettino.
“But it’s good to compete and the squad believes everyone can fight for their place in the team.
“It’s true that it’s tough and we need to be strong, but this is the rule and now we have to focus on resting and recovering to try and win the next few games.
“We are pleased in the end to get a draw because it’s important to stay alive in the competition.”
Harry Kane's 50 competitive goals for Tottenham:— Squawka Football (@Squawka) January 10, 2016
Premier League: 35
Europa League: 10
League Cup: 4
FA Cup: 1 pic.twitter.com/4SLbSDpk6U
Pochettino opted to rest his two standout performers in Kane and Dele Alli, giving opportunities to fringe players Kieran Trippier, Josh Onomah, and Kevin Wimmer.
Michel Vorm, whose late error saw Leicester end Spurs’ FA Cup hopes in last season’s fourth round, replaced Hugo Lloris in goal.
Ranieri made eight changes in total, handing a full debut to Demarai Gray, who signed from Birmingham. Jamie Vardy missed out following minor surgery on a groin problem.
Tottenham took the lead in the eighth minute when Onomah spotted Nacer Chadli’s run down the left channel and the Belgian’s deflected shot was parried by Kasper Schmeichel into Eriken’s path. The Dane kept his composure to drill into the bottom corner.
Their advantage lasted just over 10 minutes when Gray’s corner towards the near post found Wasilewski, who managed to get a run on Toby Alderweireld and find the corner with a header to score just his second goal for the club.
Okazaki’s impact was instant after replacing N’Golo Kante at half-time. In the 48th minute, tame Tottenham defending allowed the Japanese forward to weave his way into the box — selling Alderweireld with a smart step-over in the process — and his persistence was rewarded when Vorm could only block his initial shot straight back to him and he made no mistake with the second attempt.
Gray came within inches of capping a sprightly debut with a goal just after the hour when his dipping shot from 20 yards flew narrowly wide.
“He’s [Gray] had three training sessions with us and played so well,” said Ranieri.
“We have to improve our movements on the wings but he has a very good future with us.”
Pochettino sent on Kane midway through the second half to try and salvage a replay but the hosts were restricted to long-range efforts for most of the closing stages.
But then, with just two minutes of normal time remaining, referee Robert Madley adjudged Nathan Dyer to have handled as he challenged Danny Rose in the box.
Kane emphatically dispatched the penalty to bring up his half century and ensure the teams will face off for a third time this month.
Vorm 4; Trippier 6, Alderweireld 4, Wimmer 5, Rose 6; Dier 6 (Bentaleb 67 5), Carroll 5 (Kane 67 7), Eriksen 6, Onomah 6, Chadli 6 (Alli 72 5); Son 5
Lloris, Winks, Davies, Lamela,
Schmeichel 6; De Laet 7, Wasilewski 7, Benalouane 6, Chilwell 6; Inler 6, Kante 5 (Okazaki 46), King 6, Dyer 6; Ulloa 6, Gray 7 (Albrighton 64 6)
Schwarzer, Drinkwater, Morgan, Simpson, Mahrez