Kane's 49th-minute strike after Lloris had made a series of outstanding saves ensured the visitors completed a 4-0 win on aggregate.
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.
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.
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.
It is the third setback in a week for Dortmund, which was knocked out of the German Cup by Werder Bremen and then drew with Hoffenheim in the league.
Bayern took advantage of Dortmund's second successive league slip-up by beating Schalke 3-1 in the late game, lifting the six-time defending champion back to second place.
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.
Pulisic's contract with Dortmund ran until the end of next season.
Dortmund finished level on points with Atletico in Group A and sealed top spot thanks to a better goal difference.
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.
In two clashes of teams with perfect records, Barcelona beat Inter Milan 2-0 and Dortmund dismantled Atletico Madrid in a 4-0 rout.
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.
The game's only goal came in the 28th minute of the first half on Friday when the German team was handed a penalty.
The 30-year-old center back is the third new arrival since Unai Emery replaced Arsene Wenger as manager in May, joining full-back Stephan Lichtsteiner and goalkeeper Bernd Leno.
Frankfurt, which had twice come from behind to pull level, is two points behind Bayer Leverkusen, which occupies the last Champions League qualification spot.
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