Finnigan struck only six minutes after half-time at the Falkirk Stadium to put the icing on an excellent spell around the break for the hosts.
Defender Thomas Scobbie gave John Hughes’ side the breakthrough in first-half injury-time with his first goal of the campaign.
And when Finnigan nodded in the second, the home side’s possession play allowed them to toy with their visitors.
They were given a scare with 14 minutes remaining, however, when substitute Deividas Cesnauskis pulled a goal back from a tight angle.
But there was to be no equaliser from the Tynecastle men, allowing Falkirk to cut the cap between the sides to just two points.
Hearts went into this match with Gary Glen seeking a hat-trick of consecutive first-team goals as he celebrated being named the SPL’s young player of the month for April.
The 18-year-old’s winner in the 1-0 win against Inverness – the Tynecastle team’s last outing – ensured a victory against Falkirk would guarantee them seventh spot.
However, the teenager’s night was to end in less-than-happy circumstances as he limped off in the 34th minute and the night was not to get any better for the visitors.
Glen was injured in a mistimed challenge from Gerard Aafjes which had the Edinburgh side screaming for a penalty.
However, referee Stuart Dougal rightly ruled the incident had taken place outside the box and Laryea Kingston’s effort from the free-kick was blocked by Scobbie.
Falkirk defender Jack Ross paid for signing a pre-contract deal with St Mirren with his place in the squad but his replacement at right-back, youngster Chris Mitchell, impressed in his first SPL start.
Mitchell wouldn’t have had time to settle before the Bairns came close to a dream start after only three minutes.
The former Newcastle youngster evaded a couple of tackles before feeding Kevin McBride on the right flank but Scott Arfield’s shot from his fellow midfielder’s centre was deflected just wide and into the side netting.
A lightning break saw Glen and Calum Elliot combine to feed Laryea Kingston in acres of space on the right as Hearts came close to a breakthrough in the 25th minute.
However, Falkirk goalkeeper Shane Supple got down low to his left to turn the midfielder’s drive round the post.
Kingston’s corner then picked out skipper Christophe Berra eight yards out but the defender nodded the ball straight into the arms of Supple.
Pedro Moutinho had a shot tipped wide of the target by Steve Banks but there was nothing the Hearts keeper could do just before the break as Scobbie volleyed in Moutinho’s corner after Michael Higdon’s overhead kick had been touched around the post.
Hearts were eager to begin the second-half brightly in search of a quick equaliser but it was Falkirk who doubled their advantage in the 51st minute.
Banks was left hopelessly exposed by his defence when Patrick Cregg picked out Scobbie on the left side of the area.
The full-back’s header across the six-yard box found Finnigan without a marker and with the simplest of tasks of nodding home.
Hearts were denied their second penalty claim of the night in the 74th minute as Cesnauskis went down theatrically under a challenge from Mitchell.
But referee Dougal instead showed a yellow card at the Lithuanian for a dive.
The winger was not to be denied three minutes later, however, as he latched on to a rebound from Kingston’s cutback and beat Supple from almost on the goal-line.
Falkirk were in danger of allowing their lead to slip at this point as Hearts gained confidence from the strike.
However, replacement striker Jamie Mole blazed recklessly over and fellow-substitute Lee Wallace drove past as time ran out on the Edinburgh side.