Although Bristol remain at the bottom of the Premiership, the victory puts Richard Hill's men just one point behind struggling defending champions Wasps.
Woods has struggled in his last three starts after ending his long wait for his 72nd title with victory at Bay Hill in late March, but he carded solid rounds of 70 and 69 to be just one shot off the lead at the halfway stage in Dublin, Ohio.
Ireland received a straight red card following a dangerous tackle towards the end of the third quarter, but despite his dismissal and goal chances for Monaghan duo Eoghan Duffy and Keith Hill, Queen's held out for victory.