Dani Olmo's injury-time strike saved Spain in a 2-1 win against Georgia on Sunday, while France also rebounded from an opening draw in World Cup qualifying by beating Kazakhstan.
Elsewhere, England, Italy and Germany all had to work hard in the Balkans to gain their second victories in two matches.
Denmark also stayed perfect as they made a double statement, protesting the treatment of migrant workers in Qatar before putting eight goals past Moldova, while Switzerland eked out a victory over Lithuania in a game delayed because one of the goals was too big.
Spain, who had been held to an embarrassing home draw by Greece in their opening game in Group B, fell behind to a goal by Khvicha Kvaratskhelia just before half-time.
Ferran Torres of Manchester City levelled in the 56th minute with his fifth goal from just six starts.
Spain's second-half pressure finally paid off in injury time.
Olmo's rasping strike from 25 metres flew into the top corner.
"It was perhaps one of the games where I suffered the most, but, for the joy at the end, it was worth it," said Spain coach Luis Enrique.
"I'll sign to suffer if in the end we win but we saw the spectre of qualification getting very complicated."
In Tirana, Kane ended his six-game England goal drought as he inspired a 2-0 win in Albania in Group I.
Gareth Southgate's side struggled to break down the well-organised underdogs until the Tottenham striker netted late in the first half.
It was the England captain's first international goal since he scored against Kosovo 500 days ago in November 2019.
He set up England's second for Mason Mount in the second period.
"As a striker you go through spells where things don't go your way," Kane told British broadcaster ITV.
"You have to stay focused on doing your job for the team and we're grateful to get the three points."
Serge Gnabry gave Germany the lead against Romania in Bucharest after 17 minutes.
The visitors dominated much of the game, but had three narrow escapes in the closing minutes before winning 1-0.
"We should have scored the second goal, so we would not have been in danger of losing two points," said German goalkeeper Manuel Neuer, in remarks reported by FIFA.
"We are lacking coolness, cleverness, maybe a bit of will inside the box."
Italy kept their campaign on track with a 2-0 win over Bulgaria.
- 'Getting chills' -
Andrea Belotti scored the first goal from the penalty spot just before the break with Manuel Locatelli sealing the win with his first 'Azzurri' goal with eight minutes to go.
"My first Italy goal, it's incredible, I'm still getting chills," said Locatelli.
In Copenhagen, Denmark cemented their grip on first place in Group F by thrashing Moldova 8-0, with Kasper Dolberg and Mikkel Damsgaard both scoring twice.
The victory was preceded by the Danish team joining other European national sides in protesting the treatment of migrant workers in Qatar, host of the 2022 World Cup.
Danish players wore jerseys with the slogan "Football supports CHANGE" as they lined up for a photograph before the game.
The Danish Football Union said the shirts would be put up for sale and the money given to migrant workers in the Gulf state.
Kick-off in Switzerland's home game against Lithuania in St. Gallen was delayed by 15 minutes on Sunday because one of the goals was too big.
When played started, Xherdan Shaqiri scored after just two minutes as the hosts on 1-0.
Ousmane Dembele scored the opening goal as holders France won 2-0 in Kazakhstan to claim their first victory in Group D after drawing at home to Ukraine in their opener.
Sergei Maliy put through his own net for France's second goal before half-time but Kylian Mbappe had a second-half penalty saved after coming off the bench.
"People might have thought Kazakhstan would let us score five goals but no, they played well," Paul Pogba, who returned to the starting line-up, told broadcaster TF1. "We got the result and we can be happy."