Liverpool went ahead through Roberto Firmino's scuffed finish, but Norwich equalised through Dieumerci Mbokani's cheeky backheel before debutant Steven Naismith put the hosts ahead before half-time.
A Wes Hoolahan penalty gave the Canaries a two-goal cushion in the second half, but the Reds engineered an amazing turnaround through Firmino's second and strikes from Jordan Henderson and James Milner.
Sebastien Bassong's stoppage-time rocket looked to have given the hosts a point, but Lallana found the winner with almost the last kick of the game.
Both clubs provided their respective lineups ahead of kick-off, with the Reds making just one change from the side that lost 1-0 to Manchester United last time out, as Jordon Ibe replaced Lallana:
There were few early chances in the game with neither goalkeeper having much to do, but Norwich looked the more threatening side going forward.
However, it was the Reds who took the lead after 18 minutes, Firmino beating the offside trap as Milner put him through on goal, and a weak shot from the Brazilian bobbled in off the post.
Per Opta, it was the first shot on goal from either side:
Firmino almost immediately returned the favour for Milner as he put the 30-year-old through on goal, but Robbie Brady's crucial intervention kept the score at 1-0.
In the 28th minute it was 1-1, as Liverpool, once again, failed to deal properly with a corner—a common problem for them this season.
The Reds initially half-cleared the set piece, but it was headed back into the box and eventually fell to Mbokani, whose powerful backheel nutmegged Mamadou Sakho and nestled in the back of the net.
Norwich claimed the momentum after their opener and dominated possession, and they had a fairly decent penalty shout denied when Milner appeared to take down Naismith in the area.
It mattered little, though, as Naismith put Norwich ahead five minutes before half time, the former Everton man firing a powerful finish into the far corner from a tight angle having received a threaded ball from Hoolahan.
Liverpool were rocked, and they continued to look nervy after the break, and just six minutes into the second half Alberto Moreno conceded a penalty as he unnecessarily went into the back of Naismith, per AFP's Tom Williams:
Hoolahan fired home from the spot with an unconvincing strike over the diving Simon Mignolet, but it mattered not to an ecstatic Carrow Road.
The home crowd's celebrations lasted less than two minutes as Henderson fired home a terrific finish having been left in acres of space by the Norwich defence in the box.
And Firmino bagged the equaliser three minutes after the hour, the Canaries defence once again at fault as substitute Lallana found the Brazilian in space for him to clip home.
Norwich were firmly on the ropes and Liverpool were eventually gifted the knockout blow by Russell Martin.
The centre-back's woefully under-hit back-pass from the half-way line fell into the path of Milner, who raced into the box and slid a terrific finish past Declan Rudd with 15 minutes remaining.
It was the visiting fans' turn to go mad, per 97.3 Citi FM:
However, there were to be further twists in the tale, as Bassong equalised in stoppage-time, firing home from range with a left-footed finish.
The Reds were not finished yet, though, as the ball pinged around the Norwich area a couple of minutes later, eventually falling to Lallana who lashed home the winner.
Liverpool manager Jurgen Klopp said he felt his side deserved all three points following the chaotic clash, per BT Sport (via the Mirror's Joshua Evans):
I think we were the deserved winner in a spectacular game. I think it is a new record but we have to solve our problems because this will not always be possible.
Of course we were lucky, but we were not too lucky when they equalised in overtime. It is very emotional. We are in the middle of the chaos and react, sometimes we know what we are doing, sometimes we don't. The game is enough of a story.
BT Sport Football provided the comments of Milner and match-winner Lallana:
Meanwhile, Naismith bemoaned the Canaries' defensive frailties, per BBC Sport's Gary Rose: "Defensively, it was very poor, and that is what cost us. If 11 men can't defend their goal for two minutes, then you will get punished."