Scharner got 61 minutes under his belt in his first appearance in Albion colours.
Chris Wood started the scoring in the 27th minute to give the Baggies a 1-0 half-time advantage.
And the Kiwi striker made it a double when he wrapped up the game with a fourth in the third minute of stoppage time.
In between Paul Downing scored three minutes after the break and Simon Cox pressed his first-team claims with a 65th-minute strike. New recruit Somen Tchoyi boosted his fitness by playing the whole behind-closed-doors encounter at Albion's training ground.