Franzen tipped in a pass from Niklas Lidstrom 52 seconds into the third period, breaking a 3-3 tie, and the Red Wings eked out the win, which tied the 1975-76 Flyers and 1929-30 Boston Bruins for the NHL mark.
Detroit will seek to break the record at Joe Louis Arena Tuesday against Dallas.
Franzen also had an assist, while Henrik Zetterberg contributed a goal and two assists for the Western Conference-leading Red Wings.
Joey MacDonald earned the victory, making 26 stops.
Sergei Bobrovsky yielded four goals on 25 shots as the Flyers lost for the fourth time in six games.
Brayden Schenn scored twice and Max Talbot also tallied for Philadelphia.