Kane headed home from Christian Eriksen's corner in the 89th minute to make up for two glaring misses earlier in the match at Selhurst Park.
Blind, 32, has an implantable cardioverter defibrillator (ICD) in his chest to detect an irregular heart beat and deliver a life-saving shock in the event of sudden cardiac arrest.
With two defeats from their opening two games, the Danes are now only two wins from improbable glory.
A round-up of Friday's action in the UEFA Nations League.
UEFA said Eriksen, who was given CPR by medics on the pitch, was transferred to hospital where the 29-year-old was stabilised, with officials saying he was conscious. Following is a list of players who have collapsed on the pitch during a match.
Manchester United kicked off their Europa League campaign poorly, losing 0-1 at home to Real Sociedad, while Arsenal's new signing Marquinhos scored in a 2-1 win at FC Zurich on Thursday.
Manchester United's hopes of moving second in the Premier League were scuppered as Michael Olise's stunning last-gasp equaliser earned Crystal Palace a 1-1 home draw on Wednesday.
Belgium finished top of Group B with a maximum nine points, with Denmark, Finland and Russia all on three points but with the Danes ahead on goal difference.
With less than a month of the Premier League season left it remains Chelsea's title to lose but Tottenham Hotspur defiantly stayed in contention by grinding out a 1-0 victory at Crystal Palace on Wednesday.
Harry Kane's hat-trick turned Tottenham Hotspur's potentially tricky FA Cup fifth-round tie at Championship side Fulham into a Sunday stroll by the River Thames as they won 3-0 at Craven Cottage.
Manchester United manager Erik ten Hag backed Bruno Fernandes to continue his role as stand-in skipper.
Tottenham Hotspur manager Mauricio Pochettino likes to erase bad memories and cherish good ones and it is a trait he hopes his players have acquired ahead of Saturday's clash at Premier League champions Manchester City.
Images from Saturday's action in the English Premier League
Tottenham Hotspur beat local rivals Arsenal with second-half goals from prolific pair Dele Alli and Harry Kane on Sunday to maintain their Premier league title challenge.
Eriksen seeks return to 'normal life' with Brentford
The Danish international scored the winner as Spurs beat Bournemouth 1-0 in the Premier League on Saturday and Eriksen has set his sights on another good showing against the European champions at the Santiago Bernabeu.
Dele Alli condemned Champions League holders Real Madrid to a horrendous Wembley debut with two goals as Tottenham Hotspur reached the last 16 with a stunning 3-1 victory.
Gylfi Sigurdsson scored a stoppage time winner after Christian Eriksen's double as Tottenham Hotspur stormed back from 2-0 down to beat Southampton 3-2 in the Premier League on Sunday.
Kevin de Bruyne came off the bench to inspire Belgium in a 2-1 comeback win over Denmark
Spurs' triumph at the Etihad Stadium moved them within two points of Premier League leaders Leicester City and kept them in second place above Arsenal on goal difference with 12 matches to play
England forward Marcus Rashford was on target again as Manchester United earned a nervy 3-1 victory over Everton in the FA Cup third round on Friday.
The incident was one of many involving match officials being confronted in the top flight at the weekend, with the FA also charging Liverpool manager Juergen Klopp with improper conduct for his outburst at an assistant referee.
Cristiano Ronaldo came off the bench to earn Manchester United a hard-fought 2-1 victory at Everton in the Premier League on Sunday, taking his career goal tally to 700 in the process.
England's Adam Lallana scored his first international goal with the last kick of the game as they beat 10-man Slovakia 1-0 in their World Cup Group F qualifier on Sunday to get new manager Sam Allardyce's reign off to a winning start.
Manchester United climbed to third place in their quest for a berth in next season's Champions League.
Despite being without key performer Kevin De Bruyne, world number one ranked Belgium were rarely in trouble against a Russia side that struggled to impose themselves.
Kane, who became a father for the first time last Sunday, celebrated with a treble -- and it could have been more -- as Spurs coasted to a surprisingly easy win, their seventh straight victory in all competitions, to move into second in the table.
Two goals from Marcos Alonso gave Chelsea a 2-1 victory over Tottenham Hotspur on Sunday in the first-ever Premier League game at Wembley Stadium.
Headers by Romelu Lukaku and Nemanja Matic earned Manchester United an unconvincing 2-0 victory over Brighton and Hove Albion in a low-key FA Cup quarter-final played in freezing conditions on Saturday.
Cristiano Ronaldo scored on his return to the starting line-up as Manchester United sealed a place in the Europa League knock-out rounds with a 3-0 victory over Sheriff Tiraspol at Old Trafford on Thursday.
'We're going to have time to talk in the future but any big player in the world will always be welcome at PSG.'
Jose Mourinho, sacked by Manchester United last season after his reign there turned sour, has been charm personified since swapping punditry for the Tottenham job.
Liverpool manager Juergen Klopp has a midfield conundrum and his Tottenham counterpart Mauricio Pochettino must decide whether Harry Kane is ready for action as both bosses face selection dilemmas for Saturday's Champions League final.
aka was excellent in the opening minutes and it was his cross that was headed into the net by Martinelli for the opener, but he went down injured shortly afterwards after a challenge from Forest's Renan Lodi. Saka tried to continue but came off after 27 minutes.
Qatar no playground for playmakers as defenders clamp down
Tottenham Hotspur booked their place in the FA Cup semi-finals but an injury to Harry Kane marred the party as they marked their last tie at their White Hart Lane home with a 6-0 thrashing of third-tier London rivals Millwall on Sunday.
The defeat halted Brighton's impressive three-game winning streak which included a 4-1 win over Chelsea and a 3-1 victory at Arsenal in the League Cup on Wednesday.
"Harry Kane, he's one of our own," Tottenham Hotspur supporters sing in praise of the 21 year-old striker from Chingford, the London suburb where David Beckham learnt his football.
Chelsea ran riot at Swansea City, scoring four times before halftime on the way to a 5-0 win that sent Jose Mourinho's side five points clear in the Premier League on Saturday.
