First-half headers from Barney Stewart and Finn Yeats sent Falkirk into the Scottish Cup semi-finals and ended Dundee United's hopes of reaching Hampden.

The hosts made the most of a dominant opening spell and despite Julius Eskesen pulling a goal back for United, Jim Goodwin's side could not stop John McGlynn's side enjoy another step forward in what has already been a season to remember.

Falkirk started brightly and found themselves ahead after just 10 minutes. Brad Spencer swept in a shot from the edge of the box that keeper Ashley Maynard-Brewer could only parry away.

Barney Stewart was well-placed to head in the rebound and score his seventh goal in 13 games for the Bairns.

They opened up a two-goal cushion midway through the half, when Leon McCann found space on the left to deliver an inviting cross which Yeats met with a thumping header past Maynard-Brewer.

At that point, as Falkirk dominated possession and kept pushing forward, it seemed like a question of how many the hosts might score but just before half-time, United grabbed a lifeline.

Will Ferry cut the ball back across goal from the left-hand side of the box and Eskesen, who scored the winner the last time the teams met, firmly side-footed beyond Scott Bain.

Immediately after the break, United forced two corners and showed an intent that brought noisy encouragement from the travelling support as Falkirk began to look uncertain for the first time.

Bain was forced into a huge save when Zac Sapsford burst in on goal but the visitors could not make the most of a spell on top.

Johnny Russell was introduced as United pressed for an equaliser and he almost had an instant impact, producing a clever flick that allowed Ferry to cut back for Sapsford, whose finish came off the inside of the post and bounced into Bain's arms.

At the other end, Calvin Miller did likewise in a frantic finish with Falkirk seeing the game out.

Last season's Championship winners will now look to Sunday's semi-final draw to learn who they will face at the national stadium next month as they look to take another step towards silverware in..

Falkirk's season just gets better and better. For most newly-promoted sides, top-flight survival makes for a satisfying outcome but McGlynn's side now have a trip to Hampden and cup dreams to add to a near-certain top six finish.

When these sides last met at the Falkirk Stadium, a red card for the hosts played a part as United came out on top in a 3-2 thriller. That defeat clearly didn't have any affect on their confidence levels as they settled quickly and attacked with real purpose.

Stewart's opener was a goal from a man in form and his header was just the latest big contribution from a young striker with a burgeoning reputation.

But singling out individuals seems unfair when the strength has been in the collective effort of a squad that McGlynn has shaped into a very capable side.

A trip to Hampden next month will be a party for the Falkirk support, who must be loving life right now, but the team will travel confident that they can pose any team problems when on their game.

It's been 69 years since Falkirk last lifted the Scottish Cup but the Bairns might be believing this is their year.

For Dundee United, defeat looks like a season petering out before the clocks have even changed.

While a top-six place is mathematically still possible, it's almost impossible to see Falkirk letting their position slip from now. That realistically leaves the team playing for a 'best of the rest' seventh place with no Hampden day out to look forward to, a set of circumstances that might see pressure on Goodwin increase.

United have come through a busy run of fixtures after postponements disrupted their season, but now only have one win in six games, and that was a scrappy home victory over struggling St Mirren.

Next up is the Dundee derby and what will be another testing time, and then back-to-back games against Celtic and Rangers.

Falkirk manager John McGlynn tells BBC Scotland: "It feels great. I've just come out of the dressing room and the boys are buzzing.

"A lot of hard work had to go into that, a hell of a lot of hard work.

"We played really well in the first half and I thought we were on fire but United got themselves a lifeline right before half-time. From our point of view it's poor game management and they took the initiative and put us under pressure. Thankfully enough, we managed to survive.

"We're just delighted to be going to Hampden.

"The fans have been magnificent and there was a good away support that made for a good atmosphere. It kind of swung one way and another for a bit that added to it.

"I'm just glad we came off to the applause from our fans. There's a great bond and togetherness between the fans and the club.

"[Being the second-highest-ranked team in the draw] suggest that we've maybe got a chance. We'll enjoy the weekend waiting for the draw."

Dundee United manager Jim Goodwin said: "We're all bitterly disappointed to go out of the competition at this stage.

"I don't want to be making any excuses but we've had nine games in five weeks and we could have done without the St Mirren game three days before a Firday night game. Add to that we've had key players suspended but it's a really disappointing outcome but you ca't afford to start like we did.

"We wont allow the season to peter out. We have a big job to close the gap on Falkirk but we'll not give up on that.

"We'll recover from tonight but we have a big game coming up next week, a local derby that's important to our supporters.

"The fans have every right to be disappointed. It's three years I've been here and we've had relative success and they want the team to be in the top six. We're very disappointed in our league position and they have every right to voice their disappointment."

Falkirk are away to Aberdeen in the Scottish Premiership on Saturday 14 March (15:00 GMT). Dundee United make the short trip to Dens Park for the Dundee derby a day later (14:00).