Mbeumo issued a promising answer with his second score of the new Premier League campaign just before the break, before netting his third after the restart.
Wissa then bundled home to hand the hosts a comfortable lead, with Yukinari Sugawara ending Southampton's wait for a Premier League goal late on.
The hosts' lineup was most notable for the absence of striker Toney, while the immediate inclusion of the newly-signed Ramsdale marked the only change made by Saints boss Russell Martin.
The former Arsenal goalkeeper was called into action early when Mbeumo drove into Southampton's six-yard box, patiently controlling the ball before picking out Mikkel Damsgaard, who forced Ramsdale into his first save from point-blank range.
The visitors won their first corner in the seventh minute, nearly making it count when Joe Aribo clipped the woodwork after directing the ball goalwards with the small of his back.
Farewell: Ivan Toney watched Brentford’s victory before he moves to Al-Ahli in Saudi Arabia (Getty Images)
Brentford enjoyed more deep drives into the visitors' final third as the halfway point in the opening period approached. However, they lacked a clinical finish, with Wissa twice looking likeliest to break the deadlock.
A timely Sugawara intervention denied the onrushing Kevin Schade a shot, and an alert Aribo clearance prevented Wissa's effort from the ensuing scramble, over the head of Ramsdale, from opening the scoring.
Thomas Frank was forced into a change in the 38th minute when Kristoffer Ajer went down and was replaced by Mads Roerslev.
Schade was close to netting just before the interval, pinging an effort off Ramsdale's left post after some loose defending from Taylor Harwood-Bellis.
The rebound fell perfectly into the path of Mbeumo, who took full advantage with a low finish in the 43rd minute.
Brentford somehow remained in the lead at the break despite the ball falling at the unmarked Adam Armstrong's feet less than six yards out in added time.
Martin made two changes for the second half, replacing Armstrong with Cameron Archer and bringing on Mateus Fernandes for Will Smallbone, and 10 minutes later introducing Tyler Dibling for the booked Harwood-Bellis.
Ramsdale conceded his second Saints goal in the 65th minute after his side once again gave the ball away deep inside their own half. The hosts quickly worked the ball to Mathias Jensen, who picked up the assist from Mbeumo's low, 65th-minute finish.
Wissa got his name on the scoresheet four minutes later from a throw-in, flicked on by Vitaly Janelt for the DR Congo international to bundle home.
Saints rarely looked like scoring though Dibling had provided a bright spark and got a shot away as the final 10 minutes of normal time approached.
Just when it seemed Southampton were going to be shut out for a third straight top-flight contest, Dibling found himself in the build-up to their first goal of the season in the fourth minute of added time.
The 18-year-old lifted the ball to substitute Adam Lallana, whose nodded delivery enabled Sugawara to score with a bouncing effort off a post.
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