After a 1-1 draw in the first-leg of the semi-final at the Falkirk Stadium, Accies' attacking midfielder Tony Andreu scored from a Jason Scotland cut-back after 15 minutes to give the Lanarkshire side the interval advantage.
Hamilton goalkeeper Kevin Cuthbert made a couple of crucial saves as the Bairns came back strongly after the break but the home side, who had not lost in their five previous matches against Falkirk this season, held firm and might have increased their lead in the closing stages had they shown more composure.
Accies' first game against Hibs, whose manager Terry Butcher watched from the stand, will be at home on Wednesday night with the return game at Easter Road on Sunday.
Falkirk started better than they did in the first leg in midweek but some slack defending cost them a cheap goal.
Defender David McCracken, just back from injury, allowed himself to be robbed by the burly Scotland on the by-line and when the veteran striker picked out the supporting Andreu, the Frenchman screwed the ball into the far corner of the net from about eight yards, with Bairns goalkeeper Michael McGovern well beaten.
The goal turned the game towards the home side who started attacking with greater confidence.
There was a Hamilton penalty claim when a cross from Louis Longridge appeared to be blocked by the arm of Craig Sibbald inside the box but referee Kevin Clancy stood firm.
Falkirk reasserted themselves and the rest of the half swung from end to end.
On the half-hour mark Bairns' midfielder Mark Millar found striker Mark Beck with a free-kick but his diving header from 12 yards drew a fine save from Cuthbert.
It was the closest the visitors came to levelling before the break with only a blocking tackle from Falkirk left-back Stephen Kingsley thwarting Longridge.
Back came Falkirk and Cuthbert had to deal with Blair Alston's 25-yard drive from distance.
Two minutes after the restart Accies defender Ziggy Gordon had to clear a lobbed shot from Rory Loy off the line, after Cuthbert had failed to clear his lines with a punch.
In the 53rd minute, as the Bairns pressed hard for the equaliser, Cuthbert redeemed himself with a great save from Will Vaulks's guided volley.
The home side survived the corner but Falkirk had their tails up, Cuthbert making another save from Conor McGrandles's low drive with Gordon completing the clearance.
With only one goal in it, the match remained fraught for Accies.
McCracken's header from a Millar cross was deflected inches wide before a swift Accies break ended with substitute Andy Ryan, on for Ali Crawford, hitting the side-netting with only McGovern to beat.
With five minutes remaining and the game stretching further, McGovern dived to parry another strike from Ryan away from danger and in the final seconds Falkirk substitute Luke Leahy headed a Ryan shot off the line, with Accies seeing out four added minutes before taking the acclaim of their fans.Fa