Kyle scored twice in the first half before completing his treble with six minutes remaining in the second.
The result eased the Ayrshire club's relegation fears and left Falkirk languishing at the foot of the table.
Kilmarnock opted to start the match with play-maker Manuel Pascali on the bench, but their solid and aggressive style worked perfectly.
Twice in the first minute, Falkirk keeper Dani Mallo had to deal with dangerous backpasses that bobbled uncomfortably towards him as the home side tried to pen in the visitors.
After 10 minutes, Kilmarnock took a deserved lead when Kyle hammered home from nine yards after Mallo could only parry Danny Invincibile's effort.
It was only Killie's 12th league goal at Rugby Park this season and how the long-suffering home fans enjoyed it as they basked in the Ayrshire sunshine.
Falkirk were struggling to break down a tight opposition defence, but Kevin McBride went close with a 28-yard effort that Alan Combe punched to safety after 19 minutes.
Falkirk boss John Hughes promised free-flowing football prior to kick-off but it was the hosts no-nonsense style that was winning the midfield battle.
Killie were in the mood and Invincibile had a chance to double their lead in the 26th minute, but he rushed his chance from the edge of the box, and Mallo saved easily.
Kyle headed his side's second goal from six yards when he latched onto a Mehdi Taouil cross in the 37th minute.
The former Scotland international reacted quickest to the swirling cross to edge the ball beyond Mallo.
It had been 20 weeks since Kilmarnock last won at Rugby Park, but their 2-0 half-time lead was no more than they deserved as they harried and overran their opponents.
In the second minute of the second half, Falkirk broke through the home defence and only a fine save from Combe prevented Burton O'Brien's low shot from 11 yards pulling a goal back.
Combe was called on again in the 50th minute, this time with excellent double save which denied a more purposeful looking Falkirk.
Scott Arfield's powerful effort was beaten away and fell to Carl Finnigan, whose volley from 10 yards was batted away by the keeper.
The match was played in a competitive spirit, with meaty challenges the order of the day.
The home support in the 5,835 crowd felt their side should have had a penalty when Kyle was fouled, but referee Charlie Richmond awarded a free-kick on the edge of the box.
Kilmarnock could have ended the contest in the 71st minute but Invincibile fluffed his header from close range after a fine cross by Taouil.
With 15 minutes remaining, Jackie McNamara let fly from 30 yards but his effort flew just over.
It was to be his final contribution as Hughes replaced him with Scobbie, much to the anger of the travelling fans behind the goal.
As the clock ticked down, Falkirk threw everything they had at Kilmarnock but it was the home side who scored again as Kyle completed his hat-trick.
The big striker rose high to nod home a header from another first-class Taouil cross to seal victory with five minutes to play.