A sixth-minute header from Michael Bolochoweckyj gave Shire the perfect start after which close-range strikes from Paul Quinn and Jordan McKechnie secured an unassailable three-goal lead.
Stirling, who suffered their first dropped points of the campaign, hit a late consolation through Jordan White.
Albion Rovers stay second after being held to a 1-1 draw at Annan Athletic. The Coatbridge side took the lead with a David Crawford penalty in the 48th minute. Rovers were denied a second when Mark McGuigan struck the bar before Scott Davidson's fierce 85th-minute shot rescued a point.
Clyde eased into fourth following a 3-0 win over struggling Queen's Park.
Atrocious defending from the visitors allowed Keiran MacDonald to steer Clyde ahead in the 50th minute and Scott Ferguson added a second.
Queen's, who remain bottom and without a single point, suffered further anguish when James Brough netted an own goal in the 71st minute.
Peterhead's blushes were spared by a late comeback in a dramatic 2-2 draw against Elgin City.
Dennis Wyness swept home Elgin's early opener and when Craig Gunn added a second in the 77th minute, the points seemed destined for Morayshire.
However, Ryan McGeever quickly pulled one back for Peterhead before Rory McAllister's flick salvaged a point.
Montrose picked up their first point of the season courtesy of a 1-1 draw against Berwick Rangers. Paul Watson was Montrose's saviour as his 84th-minute header yielded a point after Darren Lavery had given Berwick the lead midway through the second half.