The Rangers-Spurs game was for the Walter Tull Trophy - named after a player who played for both teams and died in World War I
Rangers' summer signing Antonio Colak opened the scoring from close range after good wing play by Rabbi Matondo and a flick by James Tavernier.
But Kane curled a shot into the top corner after cutting in from the left to equalise.
And the England captain scored the winner after latching onto Son Heung-min's clever pass.
Summer signings Yves Bissouma, Richarlison, Ivan Perisic, Djed Spence, Clement Lenglet and Fraser Forster all came off the bench for Spurs in Glasgow.
Elsewhere, Wolves beat Besiktas 3-0 in Alicante after Raul Jimenez, Daniel Podence and Pedro Neto all scored in the first half.
Brentford suffered a 4-0 away defeat by Wolfsburg in Germany.
Leicester played two pre-season friendlies with different teams, winning both.
Manager Brendan Rodgers' side won 2-1 at Preston with goals from Jamie Vardy and Harvey Barnes. Meanwhile, assistant boss Chris Davies' team triumphed 3-1 at Derby with Kelechi Iheanacho, Patson Daka and Callum Wright netting.
Brighton won 2-1 at Reading. A Pascal Gross header and a long-range Solly March strike gave the Premier League side a 2-0 lead before Yakou Meite pulled one back with a penalty.
Tomas Soucek gave West Ham the lead at Luton but Sonny Bradley's last-minute goal resulted in a 1-1 draw.
Nottingham Forest lost 1-0 at Union Berlin, and a Chris Martin penalty gave Bristol City a 1-0 win at Bournemouth.
Meanwhile, Watford and Southampton drew 0-0 at Wealdstone.
-- Courtesy of BBC Sport