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Kian Harratt made a game of it with a goal on his debut for Port Vale before Mbeumo sealed the victory with a fine solo effort and late penalty.
These sides met in League One nine seasons ago but have headed in opposite directions since and now sit 62 places apart.
Bees boss Thomas Frank made nine changes from the team that beat Aston Villa in the Premier League last weekend, with Ethan Pinnock and Yoane Wissa the only survivors.
Defender Kristoffer Ajer made his first appearance since suffering a hamstring injury in October while new loan signing Jonas Lossl was given a debut in goal.
Vale's break meant they had forwards James Wilson and David Amoo back available but action at both ends of the pitch was scarce in a turgid opening 20 minutes.
The first shot came in the 21st minute from Wissa, which lacked pace but caused keeper Aidan Stone some alarm before he gathered it at the second attempt.
Five minutes later the visitors did make the breakthrough. Young Danish midfielder Mads Bidstrup, whose blond locks had been much in evidence, drove at the Vale defence before teeing up Forss for a crisp shot beyond Stone.
Chances remained scarce for the rest of the first half but were all for Brentford. Vitaly Janelt curled over while, in the final minute before the break, Stone's punch fell to Wissa, who smacked his shot against the bar, with Stone saving the follow-up from Ajer.
Vale improved in the second half and their first real opening came just after the hour mark when Ben Garrity got away down the right and his cross was met sweetly on the volley by Tom Pett only for Mads Roerslev to produce a fine block.
The Bees looked to have taken a big step towards the fourth round soon after when Janelt teed up Mbeumo to score with his first touch but four minutes later Vale were back in it, Dennis Politic crossing from the left for his fellow substitute Harratt to head in.
Suddenly Vale were on the front foot, sending cross after cross into the Brentford box.
Lewis Cass shot just past the post and David Worrall came even closer, his scrambled effort cleared by Lossl with the help of his post.
Worrall held his head in his hands, and their chance of forcing an upset receded again in the 76th minute when the Bees broke and Mbeumo curled a fine effort in off the post from the edge of the area.
The visitors' progress was then secured in the 87th minute when Mal Benning fouled substitute Ivan Toney and Mbeumo coolly rolled his penalty past Stone.
Additional reporting by PA Media.