The former Celtic and Scotland striker has joined the Bairns on a short-term deal until the end of the season after rejecting the chance to stay in the Scottish Premiership with Dundee, where he spent the first half of the campaign on loan, scoring three goals.
The 31-year-old, who signed a one-year contract extension with Celtic last summer only to see the deal terminated after the club invoked a break clause, has taken the decision to drop to League One in order to get his career back on track.
Martin Rennie's side are 21 points behind leaders Cove Rangers in sixth place but only two points off the play-off positions.
Griffiths, whose former Scotland team-mate Kenny Miller is Falkirk' s assistant manager, told his new club's website: "I'm really happy to be here. Martin and Kenny made it clear that they thought I could help them with the club's promotion push.
“For me it’s about getting back enjoying football again and helping the team push up the league.”
The former Livingston, Hibs and Wolves striker returns for a second spell at Falkirk, having spent time in the club’s youth academy in season 2002-03.
Head coach Rennie hopes the striker can score the goals that will take the club into the play-offs and back into the Championship.
He said: “I’m delighted that we’ve been able to bring Leigh to the club. He’s a very talented player who has scored goals throughout his career, and I think the timing has worked out really well for both the club and the player to come in and get playing games, and I’m sure scoring goals.”