Tottenham had been completely outplayed for the first hour, but were kept in the game by Man City's profligacy and Hugo Lloris's first-half penalty save from Ilkay Gundogan.
Manchester City cruised into the FA Cup fifth round with a 4-0 thrashing of 10-man Fulham at the Etihad.
Estonia vs Germany: Ilkay Gundogan struck twice before replacement striker Timo Werner added a third to keep Germany on course to qualify for Euro 2020.
Manchester City midfielder Ilkay Gundogan was a member of the Borussia Dortmund side that Klopp guided to the 2nd of two Bundesliga championships in 2012.
Manchester City are now 12 points clear of Manchester United and moved two points ahead of Liverpool at the top.
Germany play the Netherlands at Amsterdam Arena on October 13 before travelling to the Stade de France for their return fixture with France.
Ilkay Gundogan dazzled with two goals in Manchester City's 4-0 romp at FC Basel on Tuesday to leave the Premier League leaders on the verge of the Champions League quarter-finals.
Pep Guardiola's first Champions League match as manager of Manchester City is against German side Borussia Moenchengladbach
The craziest time of the footballing calendar has come to an end with clubs spending unprecedented amount of money before the Summer Transfer Window ended late on Wednesday night.
Ilkay Gundogan was jeered by fans during a charity match recently. He insisted on that he is happy to remain at the club and is confident he can win back the love of the fans.
Ilkay Gundogan announced in April that he would not sign a new contract with Borussia Dortmund and was therefore widely expected to leave the Bundesliga side in the summer.
He is also expected to miss Dortmund's trip to Werder Bremen on Friday August 23, but should be fit for the home match against Eintracht Frankfurt on September 1.
The German Super Cup defeat to
Arjen Robben scored a heart-stopping 89th-minute winner to earn Bayern Munich a dramatic 2-1 win over German rivals Borussia Dortmund on Saturday after a thrilling Champions League final at London's Wembley Stadium.
Germany coach Joachim Loew on Thursday called up uncapped midfielders Ilkay Gundogan and Marco Reus and eight Bayern Munich players in his squad for next week's friendly against Brazil.
Borussia Dortmund on Thursday announced they had signed Nuremberg midfielder Ilkay Guendogan on a four-year contract as the German champions begin building their squad for next season.