Simon Murray scored the only goal in the 36th minute with a fine free-kick, taking his tally to five goals in seven games.
Falkirk had an excellent opportunity to level seven minutes into the second half but Callumn Morrison's penalty kick hit the bar.
Montrose were knocked off the top after seeing their unbeaten start to the season come to an end following a 2-1 home defeat to Dumbarton.
Gregor Buchanan put the Sons ahead in the 20th minute and Kalvin Orsi, with his first goal for the club, added a second 10 minutes later.
Graham Webster halved the deficit with six minutes remaining from the penalty spot.
David Goodwillie netted a first-half brace as Clyde beat Cove Rangers 2-1 at Broadwood Stadium.
Goodwillie struck his first in the eighth minute and doubled his tally two minutes before the break.
Rory McAllister gave Cove some hope with an 84th-minute penalty but Clyde held on despite having Matthew Elsdon sent off in stoppage time.
East Fife climbed off the foot of the table at the expense of Peterhead after winning the basement battle 3-0 at Bayview Stadium.
Goals from Kevin Smith, Connor McManus from the spot and Danny Denholm wrapped up the points for the Methil men.
Alfie Agyeman bagged a brace as Kelty Hearts beat winless Stenhousemuir 4-1 to stay top of Scottish League Two.
Agyeman fired Hearts into a 13th minute lead only for Adam Brown's deflected shot to level proceedings 10 minutes later.
Joe Cardle restored Kelty’s lead 12 minutes before half-time before second half strikes from Jamie Barjonas and Agyeman secured a comfortable win.
Hearts are one point clear of Annan, who came from behind to win 3-1 at Cowdenbeath.
After Robbie Buchanan put the hosts ahead in the fourth, the visitors responded after the break with Tony Wallace bagging a double, one from the spot, either side of Iain Anderson’s goal.
Forfar missed the chance to keep pace with the top two after Craig Slater’s opener was cancelled out by Darryl Duffy’s leveller in a 1-1 draw with bottom-placed Stranraer.
Kurtis Roberts’ opener and a Dale Carrick's penalty secured Stirling Albion a 2-1 triumph over Albion Rovers, who had Charlie Reilly on the scoresheet.