Falkirk made an impressive start to the game, keeping their hosts penned deep inside their own half.
But too often they saw good moved break forward in the final third.
The opening goal came a few minutes before half time when Chelsea Raymond found Eva Rule who raced at the United defence before finishing off in style.
United Glasgow returned from the half-time interval with intent and wasted no time in pulling level.
The equaliser was the catalyst for a frantic second half which saw both sides have excellent chances to get their noses in front.
With less than 20 minutes to go Glasgow won a penalty following a handball decision against Laura Dickson.
Docherty came to Falkirk’s rescue with a terrific stop.
There was no further scoring and the game entered extra time.
In the first half the visitors took the lead when sub Fiona Gardner fired Falkirk ahead.
It was soon 3-1 when Raymond collected Tayor’s throw and found Dawson with a magnificent long range pass.
Dawson made no mistake.
United Glasgow managed to pull a goal back seven minutes from the end but Falkirk defended well in the closing minutes.
Falkirk: Docherty, Dickson, Raymond, Duffy, Taylor, Simmons, Ferguson, Rule, Gilbert, Dawson, Meaney.
Subs: Gardner, MacIntyre, McLeod.
This weekend Falkirk travel to Gleniffer Thistle.
Stenhousemuir departed the cup after finding themselves on the wrong end of a 5-1 defeat at the hands of Dunfermline Athletic.
This weekend they host Hutchison Vale.