Brentford went ahead on the half hour mark as Toney clinically converted a penalty awarded for Luis Sinisterra's challenge on Toney as part of a double VAR check at a Bees corner.
Toney beat Meslier again in spectacular fashion from another set piece 13 minutes later.
But Leeds then hit back as a Ben Mee deflection took the ball into the path of Luis Sinisterra who surged towards goal before producing a brilliant finish into the bottom left corner.
Patrick Bamford was brought on at the interval as Jack Harrison made way but Brentford went 3-1 up in the 58th minute as Toney completed his hat-trick after awful Whites defending.
United's afternoon then went from bad to worse as head coach Marsch was sent off protesting the failure to award Leeds a penalty for a challenge on Crysencio Summerville who was sent tumbling in the box.
Leeds then hit back as Marc Roca converted an excellent cross from Luke Ayling who had returned from knee surgery from the bench.
But the Bees bagged their fourth just one minute later as Bryan Mbeumo latched on to another long ball and raced clear before slotting a cool finish past Meslier.
Mbeumo was flagged offside but replays showed that Koch had flicked the ball into his path through a header and more misery awaited Leeds in the 91st minute after shocking defending from Llorente.
The Spaniard looked to drag the ball back as he tried to play out from the back but substitute Yoane Wissa picked his pocket and then advanced on Meslier before slotting home to make it 5-2.
Graham Smyth, the YEP’s chief football writer, hands out the scores from an afternoon to forget for Marsch and his Whites.
1. Illan Meslier
7 - Made big saves but was generally hung out to dry and could do nothing with the free-kick, penalty and two other goals. Slight criticism perhaps for third. Photo: Steve Bardens
2. Cody Drameh
6 - Given a difficult time once or twice by Lewis-Potter but stuck to his task well. Put one very good cross in, made some big tackles. Photo: Tom Dulat
3. Pascal Struijk
6 - Tried to get forward, made some good runs, created a little danger. Some difficult moments in his own half. Photo: Steve Bardens
4. Diego Llorente
2 - Tricky start from which he never recovered. Second half defending in particular was really poor and was entirely culpable for the fifth. Photo: Steve Bardens
5. Robin Koch
5 - Well below the standards he's set so far this season. Made a few important blocks but was given a difficult time of it. Photo: Steve Bardens
6. Tyler Adams
7 - Lots of pressing and tackling to try and keep play in the Brentford half. Among the better performers. Photo: Steve Bardens