Accies' veteran striker Jason Scotland missed a couple of good chances early in an action-packed first-half which ended with Bairns midfielder Conor McGrandles hitting the bar.
The profligacy continued after the break until Darian MacKinnon curled a shot past Falkirk keeper Michael McGovern in the 61st minute which turned the tie towards the Lanarkshire side.
However, with 10 minutes remaining Beck equalised from close range to make it all square although Accies can look forward to the return game at New Douglas Park on Sunday, with Hibernian, who finished second bottom of the Premiership, awaiting the winners in the play-off final.
Hamilton went into the game unbeaten in four games this season against Falkirk, who had ousted Queen of the South in their quarter-final play-off at the weekend, and in the early stages it was easy to see why.
Scotland should have opened the scoring for Accies in the fifth minute when Falkirk defenders Johnny Flynn and Will Vaulks ran into each other leaving the striker through on goal against McGovern but the Bairns skipper made a great block to save the day.
Minutes later, with the Falkirk defence still rattled, Scotland was sent clear by Ali Crawford and he rounded McGovern but the 35-year-old pushed the ball too wide and his angled-shot was blocked for a corner by Flynn, but it came to nothing.
McGovern then had to make a point-blank save from Crawford as Accies dominated.
The home side rode their luck again when a one-two between Accies' French midfielder Anthony Andreu and Scotland ended up with the former screwing his shot wide of the target.
There was a Falkirk penalty claim in the 33rd minute when Blair Alston's goal-bound shot appeared to be blocked by the arm of Hamilton defender Grant Gillespie but referee John Beaton was unmoved.
But as the Bairns came into the game, Beck's deflected shot from 12 yards was saved by Cuthbert at a stretch.
The drama continued all the way to the break.
Hamilton skipper Martin Canning's header from a Gillespie corner was cleared off the line by Falkirk midfielder Mark Millar before McGrandles had a header from a Liam Dick cross tipped on to the crossbar by Cuthbert before the ball was cleared to safety.
In the 53rd minute Beck headed a Millar cross straight at Cuthbert from eight yards, before the keeper made a fine save from Rory Loy's drive, and the fans began to wonder if they would see a goal.
Canning then headed a Gillespie corner past the post as the catalogue of chances increased.
But just after the hour mark MacKinnon got the breakthrough when he took a pass from Crawford at the left-hand side of the box, steadied himself and bent the ball past McGovern and in at the far corner.
A fast-paced game suddenly became frantic as Falkirk battled for the equaliser but in the 75th minute it was McGovern who had to tip Louis Longridge's powerful drive over the bar for a corner, which the home side defended.
But with 10 minutes remaining Falkirk were on level terms when Loy burst into the Accies box and cut the ball back for Beck for the easiest of finishes to set up the prospect of an exciting second-leg at the weekend, although there was time for McGovern to make crucial saves from Longridge and Accies substitute Mickael Antoine-Curier.
- Sports & Recreation
- Michael McGovern