The on-loan Everton striker headed home Chris Humphrey's near-post corner in the 18th minute of a low-key encounter in Lanarkshire.
Falkirk substitute Mark Stewart later missed an open goal after Scott Arfield had struck the bar as the visitors extended their run without scoring to five games.
Jutkiewicz had not scored since netting for Swindon at Bury in April 2007 and his breakthrough came in a match otherwise lacking in goalmouth incident.
Jim Gannon had reshuffled his strikeforce following his first domestic loss against Hibernian with Chris Humphrey and Jamie Murphy brought in to inject some pace.
But it was his defensive switch that proved more effective as Stephen Craigan helped Well keep a clean sheet after replacing Steven Saunders following four games on the bench.