After a dominant first half display, the Bees went into the break one goal to the good after Marcus Forss converted home from the left hand side.
In the second period, Vale came out with much more energy but left themselves vulnerable defensively, shown by Mbuemo punishing the hosts on the break as he slid the ball into the corner with 25 minutes to go.
The Valiants did get back into the game through a powerful header by 19-year-old loan debutant Kian Harratt, who had not long come off the bench.
However, after pushing men forward for an equaliser, Mbuemo's increased quality came to show as a well taken individual effort, followed by a calmly slotted away penalty saw the visitors go back to West London very happy.
Story of the game
In a scrappy start to the affair where the Valiants were happy to just sit in and soak up the pressure, it was Brentford that had the first opening when Finley Stevens pounced on to a loose ball inside the box, but his low tame effort was held by Aidan Stone.
Without causing too many defensive problems for Vale up to this point, six minutes later a blistering counter attack saw the Bees take the lead.
Mads Bidstrup surged forward from the center of the pitch when space opened up for him and he slid in Forss on the left hand side of the box who had the composure to smash the ball past a helpless Stone.
This was when Kristoffer Ajer was allowed to run from deep, he played in Saman Ghoddos down the right whose low cross ricocheted into the path of the German midfielder, but his first time shot was scuffed and it looped over the crossbar.
Then on the stroke of half time the visitors had another great opportunity to extend their advantage when a defensive mix up allowed Yoann Wissa have space for a shot that cannoned off the crossbar.
The follow up then fell to Ajer, but from a tight angle his powerful strike was well parried away by the recovering Stone.
At the start of the second half, Darrell Clarke's outfit came out with a new lease of life with them looking far more positive and confident on the ball.
Then moments later, the hosts created by far their best opportunity of the day when a Garrity cutback from the right was latched on to by Tom Pett, but his powerful shot was bravely blocked by Mads Bech Sorensen.
However, despite Vale's pressure, on the 65th minute mark they were again caught out on the counter attack when Janelt played substitute Mbuemo through on goal and with time on his hands, he simply slid the ball into the bottom right hand corner.
Although, four minutes later the tie was brought back into life as Vale deservedly halved the deficit.
A floated cross from the left by Dennis Politic was met by young debutant Harratt who rose highest to power his header into the bottom right hand corner.
Only a minute later the home side were then so close to levelling when Lewis Cass's drive from range arrowed narrowly past the left hand post.
This seemed to provide Vale with so much more confidence and they would have been level with 17 minutes top go if it wasn't for an excellent stop by Jonas Lossl.
A Garrity flick on from a Mal Benning cross was read by Worrall, but from point blank range his poke goalwards was diverted on to the post by Lossl.
This was a moment that Clarke's side would soon come to rue as the game was put to bed inside 76 minutes thanks to a piece of brilliance from Mbuemo, who after going on a solo run, curled the ball into the bottom left hand corner from the edge of the box.
Then with three minutes remaining the winger's hat-trick was secured from the penalty spot when Ivan Toney was fouled inside the area, and from 12 yards Mbuemo sent Stone the wrong way to seal the Bees’ spot in the fourth round draw.
Valiants can be proud of second half display
Despite struggling to get a foothold on the game and providing very limited threat when going forward in the first period, Vale looked a changed side when they came out after the break.
They returned to the pitch with much more energy and positive thinking on the ball, and for a large section of that second 45 minutes, they really gave the Bees something to be concerned about defensively.
Particularly with the introduction of Politic and through Worrall pushing higher up the pitch, Vale utilised the flanks to good effect and were whipping a fair few decent crosses into the box, something which caused panic to Brentford's back three.
However, with Vale pushing more men forward in search for that leveller, it left open spaces in their defensive third.
Then with the quality Brentford possess in attacking areas through the likes of Mbuemo, Toney and Forss, it was always likely that the home side were going to pay for it and that is exactly what happened.
Star Man- Bryan Mbuemo
Mbuemo was brought on as a substitute for Brentford on the hour mark with the score at 1-0 and it was his quality going forward that was the ultimate difference between the two sides.
With Vale looking for a way back into the game, it left room in behind their defensive line which Mbuemo was able to exploit, and with the quality that he obtains, the Frenchman proved to punish the hosts with a 21-minute hat trick.
These three goals takes his tally for the season up to seven making him the Bees current top goalscorer, as well as the fact that it makes him the first substitute to net a hat trick in the FA Cup since January 2014.