Kane had sent England into an 11th-minute lead, but Ferjani Sassi levelled the scores from the penalty spot 10 minutes before the break, and it appeared that the points would be shared in Volgograd.
The Three Lions captain came up with a stoppage-time winner, however, as Gareth Southgate's side started their campaign with three points.
There was an England XI leaked recently, and that was indeed the team that started on the night. Harry Maguire was chosen ahead of Gary Cahill as part of a three-man defence, while Jordan Henderson beat Eric Dier to the holding midfield position.
As for Tunisia, Sunderland attacker Wahbi Khazri had not played since April due to a thigh problem, but the attacker was in the XI. The same could not be said for Leicester City's Yohan Benalouane, however, with the defender named on the bench.
Lingard came close to sending England ahead inside the first three minutes, but the Manchester United attacker saw his close-range effort saved by Mouez Hassen, who then kept out a header from Maguire moments later.
England looked to have another golden chance one minute later when Lingard delivered a low cross into Sterling, but the forward turned wide of the post from close range. That said, the offside flag spared the number 10's blushes.
The Three Lions were ahead in the 11th minute, however, when Kane turned home from close range. Tunisia goalkeeper Hassen could only palm a powerful John Stones header into the box, and the England captain was on hand to convert.
Hassen had to leave the field shortly after through injury, but his replacement Farouk Ben Mustapha was equally busy, and had to keep out a long-range Henderson effort in the 18th minute.
Tunisia's first attempt of the night came in the 24th minute when Naim Sliti found space in a forward position, but his effort was always too high, before Lingard fired a good chance wide of the post down the other end.
Tunisia were having their moments as the first period continued to gather pace, with Fakhreddine Ben Youssef heading wide of the post in the 28th minute following some lacklustre England defending. Maguire then headed into the arms of Ben Mustapha in the 31st minute as opportunities continued to arrive.
Tunisia were level in the 35th minute, however, when Sassi sent a low penalty into the bottom corner after Kyle Walker had been penalised for putting an elbow into the face of Fakhreddine Ben Youssef.
VAR had a look at the decision, but it stood, and Tunisia's number 13 sent the ball past England goalkeeper Jordan Pickford.
Stones had a golden chance to send England back ahead five minutes before the break, but the Manchester City centre-back had a fresh-air shot inside the box following a period of panic by the Tunisians.
Lingard had two more opportunities before the first half ended, but the attacker fired his first effort over the crossbar before turning against the outside of the post following a bright run.
Tunisia continued to take chances from set pieces early in the second period as Kane was pulled to the ground off the ball once again, moments before Alli headed into the gloves of Ben Mustapha.
England were having plenty of the ball as the match passed the hour, but Tunisia were sitting in and defending their lines.
Maguire came close to meeting a deep cross from Henderson in the 67th minute, but the centre-back was squeezed out at the final moment as the score stayed 1-1 in Volgograd.
Marcus Rashford replaced Sterling in the 68th minute as Southgate looked to his bench for the first time, and the United forward slid Kane through moments later, only for Dylan Bronn to produce a fine challenge.
Sliti then fired wide of the post down the other end of the field, before Young lifted a free kick over the Tunisia crossbar 12 minutes from time.
Ruben Loftus-Cheek came off the England bench in the 80th minute, and the Chelsea midfielder set up Rashford in the latter stages, but the forward got his feet wrong inside the box.
Four additional minutes were added, and England stole all three points at the death when Kane nodded a Maguire flick into the back of the net from close range.
Next up for England is a clash with Panama on June 24, while Tunisia will face Belgium one day before.
TUNISIA (4-4-1-1): Hassen (Ben Mustapha 15'); Bronn, S Ben Youssef, Meriah, Maaloul; F Ben Youssef, Sassi, Skhiri, Badri; Khazri (Khalifa 85'); Sliti (Ben Amor 74')
ENGLAND (3-5-2): Pickford; Walker, Stones, Maguire; Trippier, Henderson, Lingard (Dier 90'), Alli (Loftus-Cheek 80'), Young; Sterling (Rashford 68'), Kane