Having fallen behind to an unfortunate Jan Vertonghen own goal in the first period, we restored parity after the break when Harry Kane powered home a header from Nacer Chadli’s cross.
Then came one of those ‘I was there’ moments as Alli produced an awe-inspiring goal to put us 2-1 up with six minutes remaining, with Chadli also getting in on the act with a fine strike of his own deep into stoppage time as we made it three wins out of three in all competitions over the last week.
There was just one change from our last Premier League outing against Sunderland as Heung-Min Son came in for Erik Lamela after the South Korean international impressed in the midweek FA Cup win over Leicester City and scored a spectacular opening goal.
It was relatively mild at Selhurst Park in light of the recent cold weather and there was a good atmosphere inside the stadium as we got things underway.
Jason Puncheon had the first chance of the match for the home side when he bent a free-kick over the bar from the edge of the box.
We responded well and enjoyed the bulk of the possession as Kieran Trippier forced Wayne Hennessey to tip a dangerous cross over the bar with eight minutes gone. Two minutes later, Son did brilliantly to wriggle free from two challenges in midfield and fire a low shot towards goal from distance, but it went just wide.
Hennessey was called into action again soon after when he diverted a wicked-looking cross from Kane behind for a corner and, from the resulting set-play, Toby Alderweireld headed narrowly wide.
Hugo Lloris was guilty of a mis-hit pass to gift possession to Wilfried Zaha midway through the half – the latter almost crafted a chance for Connor Wickham as a result, but we were able to clear the danger.
Having dominated the early spells it became a more even contest as the half wore on. Alli had our next chance just before the half-hour mark when he combined well with Christian Eriksen but was unable to get decent contact on his volley.
Just a minute later, though, we fell behind. Eriksen gave the ball away on the edge of our box which gave Zaha a chance to cross. Vertonghen, who had made a number of vital interventions in the moments beforehand, did his best to deflect the danger away but could only divert the ball past Lloris into his own net at the near post.
It was harsh on the balance of play but we responded brightly as, within a minute, Kane was a whisker away from turning home a dangerous ball in from Alderweireld. Alli was getting closer, too, and he forced Hennessey into a save shortly afterwards when he tried to bend one in from the corner of the box.
Yohan Cabaye for Palace had a spectacular volley blocked that looked destined to hit the target shortly afterwards before we went back on the offensive with Son and Mousa Dembele going close.
It was exciting stuff as Palace responded with Zaha blazing just over as the game ebbed and flowed in pulsating fashion.
Then, with five minutes of the half remaining, Danny Rose volleyed straight at Hennessey from a brilliant Eriksen cross in our clearest opening of the half.
It was chance-a-minute stuff as Son then failed to get on the end of a lethal Kane cross with Palace clinging to their lead.
Zaha was a constant menace for them, though, and gave them a dangerous outlet on the break which they exploited well.
Five minutes after the restart we had appeals for a penalty turned down for handball against Scott Dann, although replays showed the ball seemed to hit his face.
A minute later Eriksen tried his luck from distance but his effort dipped just over the bar.
Dembele was seeing plenty of the ball in front of the back four and looked to be positive whenever possible.
With 10 minutes of the half gone, Eric Dier picked out Kane with a stunning ball forward from deep but the forward struggled to get any conviction on his header and the goalkeeper gathered easily. It was Dier’s last action as Chadli replaced him soon afterwards – a substitution that would certainly pay dividends!
Hennessey was called into action again by Eriksen on the hour mark, this time following a long-range free-kick, but he saved low to his left.
It was one-way traffic at that point and sure enough we got our leveller on 63 minutes. Chadli stood up a cross from the left and Kane rose brilliantly to thump a header past Hennessey.
There was no celebration, though, as the players grabbed the ball from the net and hared back to the centre circle, determined to get things underway again and find another goal.
Palace appeared to weather the storm, though, and five minutes later Zaha brought the best out of Lloris with a curling effort from the edge of the box. Then, from the resulting corner, Lloris produced another stunning save to deny Pape Souare, the rebound fell square into the path of Damien Delaney who looked certain to score, but Alderweireld crucially blocked to prevent the hosts from re-taking the lead.
Unfortunately, with 15 minutes to go we suffered a blow as Vertonghen hobbled off with an injury, meaning Kevin Wimmer stepped off the bench to make his Premier League debut.
Palace were a whisker away from regaining the lead with 11 minutes to go when they twice rattled the bar from a set-play. Dann’s header looped over Lloris but rebounded off the crossbar and, on the follow-up, substitute Mile Jedinak also hit the woodwork.
It was a let-off for us, but we took things into our own hands with six minutes remaining with a quite remarkable goal and surely one of the best scored by anyone in a Spurs shirt in the modern era.
Eriksen nodded the ball down to Alli just outside the box after Kane’s cross from the right, the midfielder took a touch, flicked the ball up over his own head, turned and struck a fine volley low into the bottom corner.
We still had to concentrate, though, as Lloris was alert to keep out a header from Palace substitute Fraizer Campbell, while at the other end, Alli advanced in from the left side and beat the keeper with two of the five additional minutes played – only to see his effort hit the bar.
Nabil Bentaleb, on as a late substitute for Eriksen, invaded the box from a similar position just afterwards and tried to find the top corner, but his shot zipped just over.
We finally found the third goal in the fifth minute of stoppage time as Chadli collected the ball in midfield, roamed towards the box and let fly from 22 yards – his effort curling low into the bottom corner to complete a memorable comeback win.
Crystal Palace (4-2-3-1): Hennessey, Ward, Dann, Delaney (c), Souare; Ledley, McArthur (Jedinak 78); Puncheon, Cabaye (Campbell 85), Zaha; Wickham. Substitutes (not used): Speroni, Lee, Mutch, Chamakh, Kelly.
Spurs (4-2-3-1): Lloris (c), Trippier, Alderweireld, Vertonghen (Wimmer 75), Rose; Dembele, Dier (Chadli 56); Eriksen (Bentaleb 88), Alli, Son; Kane. Substitutes (not used): Vorm, Davies, Carroll, Lamela.
Goals: Crystal Palace – Vertonghen OG 30. Spurs – Kane 63, Alli 84, Chadli 90+5.
Yellow cards: Crystal Palace – McArthur 60, Jedinak 90+3. Spurs – Trippier 79, Kane 90+3.
Referee: Martin Atkinson.