The only goal of the second leg at New Douglas Park came from Andreu (15) who capitalised on a mistake from David McCracken to tap in from close range after Jason Scotland had cut the ball back.
The tie was level following a 1-1 draw in the first leg at Falkirk and the visitors started the game well with Mark Millar and Stephen Kingsley both testing home goalkeeper Kevin Cuthbert with early efforts.
But Hamilton opened the scoring, against the run of play, when McCracken's error allowed Scotland in and the veteran striker picked out Andreu who screwed the ball into the far corner of the net from eight yards.
Cuthbert maintained his side's lead with a good save to keep out a diving header from Falkirk striker Mark Beck.
Louis Longridge and Scotland should then have added to Hamilton's advantage but both players had shots blocked just before the break.
Falkirk went close just after the interval when Cuthbert spilled the ball but Rory Loy's shot was cleared off the line by Accies defender Ziggy Gordon.
Cuthbert made up for his mistake with a brilliant point-blank stop to deny Will Vaulks and a superb save to keep out a long-range effort from Conor McGrandles.
Andy Ryan had two late chances to seal the win but he dragged his first effort wide and was then denied by a great save from Michael McGovern.
Falkirk pressed hard but Hamilton hung on to face Hibernian in the first of a two-legged final at New Douglas Park on Wednesday, while the second leg is on May 25 at Easter Road.
Elsewhere, Cowdenbeath beat Dunfermline 3-0 in the Scottish Championship play-off final to seal a 4-1 aggregate win and stay in the division next season.
Stirling Albion beat East Fife 2-0 with goals from Jordan White and Sandy Cunningham to secure a 3-2 aggregate Scottish League One play-off final win.