Bayern secure comfortable win
In other matches, Hamburger SV won 2-1 in Mainz on two goals from Son Heung-min, Hoffenheim drew 2-2 in Wolfsburg, Fortuna Duesseldorf and Werder Bremen also drew 2-2, while in the late match Schalke hosted Bayer Leverkusen.
Bayern faces Wolfsburg in the German Cup semifinals on Tuesday and coach Jupp Heynckes rested nine of his usual 11 starters. But the new Bundesliga champion, which sealed the title last week, dominated the Bavarian derby from the start and led 3-0 by halftime to end Nuremberg's run of nine matches without a defeat.
Mourinho caused a stir by showing up in Fuerth to watch the outgoing Bundesliga champion strengthen its grip on second place by routing a team heading for relegation. Dortmund hosts Real Madrid on April 24, while Bayern plays the other Spanish giant, Barcelona, in Munich a day earlier.
Mario Goetze and Ilkay Guendogan scored a pair each and Jakub Blaszczykowski added one before halftime as local fans snapped photographs of Mourinho. After Edgar Prib pulled one back for the home side, Robert Lewandowski completed the big victory by scoring in his 11th straight game for a league-high total of 22 goals.
In Munich, Bayern did not take long to take control. Defender Jerome Boateng had all the time in the world to set himself up perfectly for a first-time volley after a corner and drilled the ball in with his left foot in the fifth minute.
In the 17th, Boateng released Mario Gomez, who slotted inside the post. In the 24th, defender Rafinha scored after a well-executed Bayern move.
Rafinha gave away a penalty early in the second half, but Tom Starke read Timmy Simons' shot from the spot and saved it with his head. Starke is usually back-up to Manuel Neuer.
Xherdan Shaqiri completed the win in the 56th, before being replaced by Pierre Emile Hoejbjerg, a 17-year-old Dane who became Bayern's youngest player to make a Bundesliga debut.
''It was not so easy as it looked,'' said Franck Ribery, Bayern's captain for the match. He and veteran defender Daniel van Buyten were the only regulars to play.
With five matches to go, Bayern remains 20 points ahead of Dortmund and still has only one loss.
Hamburg ended a three-match losing streak and revived its hopes of clinching a place in the Europa League next season on a pair of goals by Son. Shawn Parker scored late for Mainz, but his team fell behind Hamburg to ninth.
Hoffenheim squandered a chance to give itself a bigger boost in the struggle against relegation. After conceding an early goal by Maximilian Arnold, his first for Bremen in the Bundesliga, Hoffenheim went ahead on a penalty goal by Sejad Salihovic and Andreas Beck, before Naldo got a late equalizer for Wolfsburg in the 86th.
In Duesseldorf, the home side led twice, with both goals coming from Stefan Reisinger, but an own goal from Martin Latka earned Bremen a draw. Zlatko Junuzovic also scored from Bremen.