Dortmund, who host Shakhtar Donetsk on Tuesday for a spot in the Champions League last eight, consolidated second spot on 46 points with third-placed Bayer Leverkusen, who scored twice in the last eight minutes to beat VfB Stuttgart 2-1, a point behind.
Lewandowski, on the pitch only after he had his three-game suspension for a red card reduced to two matches late on Friday, fired the hosts into an eighth-minute lead from close range after a cutback from Kevin Grosskreutz.
Klopp, who apologised before the game to Bayern coach Jupp Heynckes for saying the leaders had copied Dortmund's style this season, deployed Lewandowski in an unusual position behind forward Julian Schieber.
The Poland forward, linked with a transfer to Bayern after refusing to extend his contract with the champions beyond 2014, was unaffected by the switch and added another in the 20th to make it a joint league-leading 16 goals for the season.
"If there was any need for proof that people can perfrom well despite the headlines then Robert did exactly that today," Klopp told reporters. "It was a very strong performance.
"We had a really good start to the game and were in good form. "After we scored to go 3-1 up that was it. It was a great offensive spectacle."
Hannover, playing without injured midfielder Szabolcs Huszti and top striker Mame Diouf, pulled one back through Mohammed Abdellaoue just before the break but Schieber killed any hopes of a comeback when he threaded in from 15 metres for his first goal of the campaign.
Leverkusen needed two late goals to come from a goal down and beat visiting Stuttgart. Stefan Kiessling's 82nd minute penalty cancelled out Stuttgart's first half lead through Vedad Ibisevic's spot-kick.
Kiessling, who joined Lewandowski on 16 goals this season, almost got another one but keeper Sven Ulreich punched away his effort only to see Lars Bender nod in the winner from the rebound in the 86th, leaving Leverkusen seven points ahead of fourth-placed Eintracht Frankfurt.
Schalke 04 teenager Julian Draxler scored twice and set up another goal as the Gelsenkirchen club continued their domestic recovery with a 4-1 demolition of VfL Wolfsburg for their second straight win.
Germany international Draxler struck either side of an Ivica Olic equaliser and then provided an assist to Peruvian Jefferson Farfan. Klaas-Jan Huntelaar drilled in from a tight angle in the 86th minute to help Schalke move up to sixth place on 36 points.
"We were superb throughout the entire game and deserved to win by this margin," said a relieved Schalke coach Jens Keller, who took over in December and had been sharply criticised for weeks.
Schalke, level on points with fifth-placed Freiburg who drew 1-1 at Nuremberg, face Galatasaray in the Champions League later this month.
Lowly Augsburg, who occupy the relegation play-off spot, continued their improvement with a 1-0 victory at Werder Bremen.
Tobias Werner got the only goal on 29 minutes. The Bavarian club have won two in a row and have lost only one of th
