Blair Alston was the Scottish Cup hero for Falkirk as they booked another Hampden semi-final date.
Alston had fired Steven Pressley's men into a half-time lead, before a stunning second with 20 minutes left looked to have won the game.
Andy Ryan hauled Hamilton back into the game minutes later, but was kicking himself after blowing a golden chance to level with 10 minutes left.
Jonathan Page was sent off as an Accies onslaught failed to earn a replay.
Falkirk were first to threaten in front of a healthy travelling support at New Douglas Park, Stewart Murdoch seeing his shot deflected over, but after that Hamilton took charge of the early exchanges.
Jonathon Routledge swung a shot just wide in the 13th minute after a sweeping move involving Stevie May and Alex Neil, then a blunder by Jonathan Flynn gifted Gary Fisher a golden opportunity but he fired straight at goalkeeper Michael McGovern.
Accies were hunting in packs and Pressley's men were struggling to find space, but as the game approached the half-hour mark the Bairns started to come more into it, as Alston and David Weatherston tested Kevin Cuthbert.
Accies cut Falkirk open in 36 minutes. Crafty one-touch football gave Darian McKinnon the chance, but the shot in the end was too easy for McGovern. Fisher than fired wide a minute later as Billy Reid's men stayed in control of the game.
Falkirk were looking dangerous from set-pieces, and Thomas Grant's free-kick found the head of Darren Dods who headed just wide.
Hamilton thought they should have had a penalty when Routledge went down under a challenge from Dods, but referee Willie Collum awarded the corner.
Accies were trying to take advantage while Falkirk reshuffled, with Liam Dick on for Stephen Kingsley, and Louis Longbridge curled a 22 yard shot just wide but, despite their dominance, it was Falkirk who broke the deadlock two minutes before half-time.
A great counter attacking move saw Lyle Taylor pick out Grant on the right hand side, and he presented Alston with an easy chance which he duly tucked away from two yards.
In front of 3,452 fans the home side kept up a high tempo in the second half, camping around the Falkirk box, but a well disciplined back line kept them at bay.
Kevin Cuthbert had to be alert though to keep them in the game in the 53rd minute. Taylor was given freedom to run forward, and he cracked a laser off the bar. The rebound bounced down to Sean Higgins who volleyed towards goal, but Cuthbert had raced back to deny him.
Falkirk looked to have the game won with 20 minutes left from a goal fit to win any cup tie. A corner from the left was headed away by Martin Canning, but Alston thumped a simply brilliant shot right into the top corner that very few goalkeepers would have got near.
Hamilton were not dead and buried yet, though, and three minutes later substitute Andy Ryan gave them a lifeline. Fisher's shot was parried by McGovern, and Ryan stole in to take the ball away from the goalkeeper, then fired in from a tight angle which Dick tried and failed to keep out.
Ryan had given them a lifeline, but he passed up the chance to be the hero with 10 minutes to go. May controlled well on the right, and his low ball was begging to be finished, but from two yards Ryan somehow put his shot over the bar.
Accies appealed for another penalty in the dying minutes, this time Ryan hitting the deck, but again Collum gave the corner.
Page was sent off in stoppage time as frustration got the better of the home side, a vicious two-footed challenge on Taylor giving the referee no option but to order him off.
So Falkirk, having reached a Scottish Communities League semi-final against Celtic last year, head back to Hampden to continue their journey in the Scottish Cup.