Inverness will point to the stupidity of Jake Mulraney as a major contributory factor to their defeat, with the Irish winger sent off for two yellow-card offences in a three-minute spell just before half-time.
Caley Thistle barely had time to adapt to playing with ten men when Joe Chalmers brought down Louis Longridge in the box early in the second half and Muirhead slotted the spot-kick into the right corner.
Falkirk won a second penalty when Colin Seedorf tripped Tommy Robson and this time skipper Muirhead slammed his effort down the middle as Mark Ridgers dived to his left.
The visitors’ 16-year-old substitutes Jack Brown and Daniel MacKay combined on 84 minutes, with MacKay finishing brilliantly. But Falkirk made sure of the win when Watford loanee Alex Jakubiak crossed for Andrew Nelson to tap in the third goal from close range.