Following a 2-0 win in London on Feb. 19, Bayern increased its lead in the home-and-home series when Schweinsteiger scored off Franck Ribery's cross.
Lukas Podolski got a goal against his former team two minutes later, dispossessing Philipp Lahm in the penalty area and smashing the ball past goalkeeper Manuel Neuer into the roof of the net from an acute angle.
Arsenal goalkeeper Lukasz Fabianski saved Thomas Mueller's penalty kick in second-half injury time after Laurent Koscielny pulled down Arjen Robben.
In the night's other match, Diego Costa scored twice to lead Atletico Madrid over visiting AC Milan 4-1 for a 5-1 aggregate win.
Diego Costa scored in the third and 85th minutes for Atletico, which reached the quarterfinals for the first time since 1997. Atletico also got goals on Arda Turan's volley in the 40th and Raul Garcia's header in the 71st.
Kaka scored in the 27th for Milan. Italy does not have a team among the final eight in Europe's top club competition.
In Wednesday's games, Barcelona takes a 2-0 lead into its game against visiting Manchester City and Paris Saint-Germain has a 4-0 lead at home against Bayer Leverkusen.