Dortmund hit six and Bayern four in easy wins
Ilkay Gundogen and Mario Gotze each netted twice for second-placed Dortmund against bottom club Furth, who remained winless at home in their first Bundesliga season.
Poles Jakub Blaszczykowski and Roberto Lewandowski, on target for the 11th league game in a row, completed the scoring for Borussia with Jose Mourinho, coach of Champions League semi-final opponents Real Madrid, watching from the stands.
Jerome Boateng, Mario Gomez, Rafinha and Xherdan Shaqiri shared the goals as Bayern, who wrapped up the title last week with six games to spare, brought Nuremberg's nine-match unbeaten run to a halt.
Reserve goalkeeper Tom Starke, one of nine changes from the Bayern team that beat Juventus 2-0 in the Champions League on Wednesday, completed a perfect day for the Bavarians with an unorthodox penalty save from Timmy Simons.
Starke knew little about it as he dived the right way and the ball struck his head and bounced away.
Schalke 04 scored twice in the last 20 minutes to force a 2-2 draw at home to Bayer Leverkusen in a controversial clash between two sides battling for a place in next year's Champions League.
The hosts were angry when Simon Rolfes headed the visitors ahead in the 39th minute with Schalke defender Marica lying injured in the penalty area and Leverkusen were annoyed about the 86th-minute penalty which led to Raffael's equaliser.
Stefan Kiessling had put Leverkusen two ahead before Temu Puuki pulled one back.
South Korea's Son Heung-Min scored twice to give Hamburg SV a 2-1 win at Mainz 05 in a match between two teams chasing places in European football next season.
Hamburg had lost their last three games and appeared to heed the warning of coach Thorsten Fink, who told his players last week that anyone who did not pull their weight would leave at the end of the season.
Twenty-year-old Son put Hamburg ahead after the Mainz defence got themselves in a tangle just after the hour and struck again 20 minutes later with a run from the halfway line.
Shawn Parker replied for Mainz after Hamburg's defence also got tied in knots. Hamburg moved up to eighth with 41 points with Mainz one place below on 39.
In the lower half of the table, a late goal by Naldo gave VfL Wolfsburg a 2-2 draw at home to relegation-threatened Hoffenheim.
Eighteen-year-old Maximilian Arnold volleyed Wolfsburg in front before Sejad Salihovic, with a penalty, and Andreas Beck turned the game around for Hoffenheim, who stayed in 16th, the relegation play-off place, with 24 points.
Fortuna Dusseldorf, six points ahead of Hoffenheim in 15th, drew against Werder Bremen.
Bayern, who have won 25 of their 29 games, have 78 points at the top, 20 ahead of Dortmund, who are close to guaranteeing a Champions League group stage place next season.
Bayer Leverkusen, third with 50 points, are favourites to take Germany's third and final automatic group stage place after staying four points clear of fourth-placed Schalke, who are in the Champions League play-off spot.
Gotze set Dortmund on their way in the 12th minute and Gundogen turned in Blaszczykowski's pass three minutes later. The Pole headed the third in the 29th minute and then set up the fourth for Gundogen to score with a curling shot.
Gotze made it 5-0 on the stroke of half-time and, although Edgar Prib pulled one back for Furth in the 71st minute, Lewandowski added a sixth for Dortmund with 10 minutes left.
Bayern also looked unstoppable as Boateng, Gomez and Rafinha put them 3-0 up after only 24 minutes at Nuremberg, though they eased off with Shaqiri adding a goal early in the second half.
"We threw the game away in the first half," said Nuremberg coach Michael Wiesinger. "We have to apologise for the first half. Nobody played with 100 percent today."