Cairney opened the scoring just after the half-hour mark after a QPR attempted counter broke down.
The 27-year-old surged forward, the opposition defence backing off him, before slotting the ball coolly into the bottom corner from 25 yards.
QPR would go back into the dressing rooms just one goal down, however, as Luongo volleyed in from Matt Smith's header across goal in first-half injury time.
With just nine minutes left, Odoi tried to control the ball 30 yards from goal but a poor touch saw him dispossessed by Wszolek.
The Polish winger duly charged through on goal and coolly curled the ball into the bottom corner to level the game for QPR and give the visitors a share of the spoils.
FULHAM (4-3-3): Bettinelli; Fredericks, Ream, Odoi, Targett; McDonald, Cairney (Fonte 79'), Johansen; Piazon (Ayite 69'), Mitrovic, Sessegnon (Ojo 79')
QUEENS PARK RANGERS (4-4-2): Smithies; Furlong, Onuoha, Lynch, Bidwell; Manning (Cousins 74'), Luongo, Freeman, Wszolek; Smith (Washington 69'), Eze (Smyth 69')