The Borussia Dortmund forward will join the German and European champions at the end of the season after penning a five-year deal on Saturday.
Bayern club chairman Rummenigge, a prolific goalscorer during his own playing career, believes the marquee signing will help inspire Pep Guardiola's men to future glories.
"We’re very pleased about completing this transfer," he said. "Robert Lewandowski is one of the best strikers in the world.
"He will strengthen the Bayern squad and give us another boost.
"We’re delighted both parties have today signed a five-year contract until 2019."
Poland international Lewandowski joined Dortmund from Lech Poznan in 2010 and has made 115 Bundesliga appearances, scoring 65 goals.
The 25-year-old was part of the side that won the Bundesliga in 2010-11 and 2011-12 and starred in Dortmund's run to last season's UEFA Champions League final at Wembley, where they lost 2-1 to Bayern.
