Bale, who has had a difficult start to the season at Real Madrid, scored during first-half injury time after Nikola Vlasic's opener for Zlatko Dalic's men.
Stanislav Cherchesov's Russia, who reached the World Cup quarter-finals on home soil last year, only had to avoid defeat in Nicosia to join Belgium in qualifying from Group I.
Brazil started well, with Liverpool forward Roberto Firmino firing just wide in the third minute after a smart turn near the penalty spot.
French striker Olivier Giroud says the 2018 world champions "want to put that right (the defeat), as a matter of pride because of what happened over there".
In the other matches, Qatar SC beat Al Sailiya 4-0 while Al Arabi and Al Khor played out a 1-1 draw.
Roberto Mancini's side joined Belgium in the draw for next year's finals.
The Danes moved up to 12 points alongside group leaders Ireland, who drew 0-0 away to Georgia.
The 67-year-old Ranieri takes over from Eusebio di Francesco, who was sacked after a poor start to the season.
Al Rayyan blank Al Shahania 3-0, Al Ahli beat Umm Salal 2-0.
World champions France claimed a 1-0 win away to Iceland.
The veteran star hit his 94th goal for the Portuguese and took his overall career tally to 699.
The 16-year-old admitted breaching Football Association rules and must also complete a face-to-face education course and pay a £350 fine.
Wayne Rooney's wife Coleen had accused Rebekah of leaking stories
Sayed Aghazada was found guilty of failing to "report and prevent" former federation president Keramuudin Karim from carrying out repeated acts of sexual assault between 2013 and 2018.
The 30-year-old has spent his entire professional career at Bayern, but will "think things over" if Kovac only sees him as a replacement.