In conditions over 30 degrees, the Bees more than matched their higher level opponents in the first half.
Shaq Coulthirst had some of the best opportunities,firing narrowly wide after neat link up play between Jack Taylor and Elliott Johnson, making his return from injury after almost 11 months out.
Further neat build up play through the Barnet midfield led to another chance for Coulthirst, this time the forward saw his effort saved.
Salomon Rondon had an effort cleared off the line by Craig Robson before Matt Phillips curled an effort wide of the post.
Both sides rung the changes at the break and West Brom took the lead 10 minutes after the restart. Kyle Edwards with an excellent finished past the helpless Trialist GK in the Barnet net.
Hal Robson-Kanu doubled the Baggies advantage from the penalty spot before he doubled his personal tally with a good effort from the edge of the box.
Just before the end, Mason-Clark turned and shot from 25-yards out but Brad House was equal to it in the West Brom goal.
Barnet: Cousins (GK), Alexander, Johnson, Reynolds, Robson, Sweeney, Kyei, Boucaud, Sparkes, J. Taylor, Coulthirst.
Barnet subs: Fonguck, Trialist One, Adams, Barham, Payne, Pascal, Mason-Clark, Tutonda, Syla, Shomotun, Sule.
Referee: Josh Smith