The Wizards team showed 11 changes from Sunday's starting line-up and a magnificent defence helped them to victory.
The home side got on the scoreboard first when full back James Stephenson kicked a penalty in the first quarter. The Wizards hit back with tries from centre Richard Thomas and left wing Steve Davies.
Outside half Jamie Davies, who is in the Barbarians squad to face Richmond in 12 days time, converted both to give the visitors a 14-3 lead at the interval.
In the second half the Quins fought back resulting in a pushover try by Number 8 Chris Jones. Davies and Stephenson then traded penalties before Davies landed two more kicks to complete the game.