Julian Brandt and Nico Schulz have joined from Bayer Leverkusen and Hoffenheim, respectively, while Belgium forward Thorgan Hazard has switched from Borussia Mönchengladbach.
Bayern Munich crushed Eintracht Frankfurt 5-1 to lift their seventh successive Bundesliga title.
Mats Hummels and Robert Lewandowski set Bayern back on track toward a record-extending seventh straight title.
Just when it seemed Dortmund's title chances would take a serious knock in heavy rain, Jadon Sancho set up Reus to score in the 93rd minute.
Both Bayern Munich and Dortmund have 57 points after 25 of 34 rounds with Bayern top on goal difference.
Ji, who endured a frustrating half season at Dortmund in 2014, struck in each half to hand Lucien Favre's side only its second defeat of the season and open the door for Bayern Munich to move level on points.
Rafinha, who hasn't played since Bayern's 3-1 loss at Bayer Leverkusen on Feb. 2, made his displeasure known publicly after the 1-0 win over Hertha Berlin.
Bayern Munich, which trailed by nine points at one stage, is currently tied on points with Dortmund but remained second due to the leaders' better goal difference.
Nuremberg goalkeeper Christian Mathenia made a host of saves to deny Dortmund a first win in three league games.
Pizarro, aged 40 years and 136 days, broke the previous record held by Bremen midfielder Miroslav "Mirko" Votava.
The home side got off to a flying start and scored after 13 seconds as Philipp Max's cross was deflected in by Bayern's Leon Goretzka.
Jadon Sancho starred for Dortmund in the absence of injured captain Marco Reus, but instead of opening a 10-point gap ahead of Bayern's home game against Schalke later, they allowed Hoffenheim to come back.
Borussia Dortmund are already seven points clear this season and looks a far better bet for the title than Niko Kovac's disjointed team.
Bayern, the six-time defending champion, did not give the extent of Neuer's injury, nor an indication of how many games he will miss.
The 25-year-old Alcacer joined the German leaders on a season-long loan in August, and Dortmund had the option of making the deal permanent.
Bayern fell seven points behind Dortmund, which remains the only unbeaten team in the Bundesliga.
Dortmund, which had a four-point lead, moved seven points clear of Bayern and stayed four clear of second-placed Borussia Moenchengladbach, which defeated Werder Bremen 3-1 away.
Bayern started the season with seven wins across all competitions but have dropped to fifth after seven rounds of the league, four points behind Dortmund.
Bayern were on a roll after starting the season with seven wins in all competitions, but the team dropped points with a 1-1 draw against Augsburg on Tuesday before losing at Hertha Berlin 2-0 on Friday.
The Spanish striker, who joined Dortmund from Barcelona in the offseason, scored the winner to give Lucien Favre's men three vital points in the league.
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