Götze provided an assist and netted a goal in Dortmund's 3–0 second leg defeat of Shakhtar Donetsk on 5 March 2013, completing a 5–2 aggregate victory over the Ukrainian champions as Dortmund progressed to the quarter-finals of the Champions League. The result meant that Dortmund had advanced to the quarter-finals for the first time in 15 years. Götze was ruled out of the final against Bayern Munich at Wembley Stadium, London, as he suffered thigh injury against Real Madrid in the second leg of the semi-final, which his side lost 2–0 but still progressed to the final through an aggregate score of 4–3. His Dortmund side lost the final 2–1 after a late Arjen Robben goal sealed Bayern's victory.
On 23 April 2013, it was announced that Götze was moving on 1 July 2013 to rivals Bayern Munich after the team had triggered Götze's release clause of €37 million. The transfer made Götze the most expensive German player of all time. Mesut Özil eventually broke the record later in the summer, transferring to Arsenal for €50 million. Dortmund coach Jürgen Klopp claimed that the reason behind Götze's transfer to Bayern was the playmaker's wish to play under manager Pep Guardiola, formerly of Barcelona. Klopp admitted his annoyance at the timing of the announcement of Götze's move, as it was barely 36 hours before Dortmund's Champions League semi-final with Real Madrid. Klopp later said that Dortmund had no chance of convincing Götze to stay with Dortmund, as he claimed, "He [Götze] is a Pep Guardiola favourite."
On 11 August 2013, Götze made his Bayern Munich debut, coming on as a substitute for Mitchell Weiser in the 60th minute. Götze netted two goals to help Bayern get a 1–4 win against Hungarian champions Győri ETO in a friendly match. He made his league debut for the club in a Bundesliga home match against 1. FC Nürnberg on 24 August, which Bayern won 2–0. On 19 October, Götze came on as a substitute and made two assists to help Bayern from a one-goal deficit to get a 4–1 win against 1. FSV Mainz 05.
On 23 October, Götze scored his first competitive goal for Bayern in a 5–0 Champions League group stage win over Viktoria Plzeň at the Allianz Arena. He also assisted Bastian Schweinsteiger in the game. On 26 October 2013, Götze came on as a substitute for Toni Kroos in 25th minute and scored his first Bundesliga goal for Bayern Munich with a header in a 3–2 win against Hertha BSC. On 2 November, he made his return to starting line-up in a 1–2 away win for Bayern against TSG 1899 Hoffenheim. On 23 November, Götze came off the bench and scored the first goal of a 0–3 away victory over former club Borussia Dortmund. He did not celebrate the goal out of respect to Dortmund. In the team's next fixture against CSKA Moscow, Götze scored the second goal of 3–1 away win in the 2013–14 UEFA Champions League group stage. On 7 December, Götze helped Bayern by netting a goal in the closing minute, as well as assisting Thomas Müller, in a dominant 7–0 away victory over Werder Bremen. On 17 December, Götze scored a long-range goal in the 47th minute against the AFC champion Guangzhou Evergrande, helping Bayern win 3–0 and advance to 2013 FIFA Club World Cup Final, where he came on as an 80th-minute substitute. Bayern won the match 2–0.