ALBION created the better chances but were unable to make them count in an FA Cup stalemate that means the two sides will have to do it all again in 11 days' time at The Hawthorns.
Both clubs will go into the hat for tomorrow’s fourth-round draw – but the Hoops were fortunate to still be in the competition.
Julio Cesar was by far the busier goalkeeper as he pulled off a string of fine saves and Romelu Lukaku twice went close to opening the scoring.
Shane Long looked to have won it for the visitors with his 79th-minute strike but Kieron Dyer pounced in the first minute of stoppage time to net an unlikely equaliser.
Prior to that the Baggies also had to endure the disappointment of losing Zoltan Gera and his replacement, Marc-Antoine Fortune, to injury before half-time.
Steve Clarke made three changes to his team, who took to the field wearing black armbands in honour of legendary Baggies striker Derek Kevan, who died yesterday, aged 77.
Boaz Myhill, George Thorne and Long came in, with the latter partnering Lukaku up front in a switch to a 4-4-2 formation.
The men to make way were Youssouf Mulumbu, who did not travel with the squad as he prepared to join up with DR Congo for African Nations Cup duty, and benched duo Ben Foster and Fortune.
Barclays Premier League strugglers Rangers made four changes to the team that beat Chelsea 1-0 in midweek.
Dyer, DJ Campbell, Ji-Sung Park and Tal Ben Haim replaced Fabio da Silva, Shaun Derry, Nedum Onuoha and Junior Hoilett.
The first genuine effort on goal came in the 14th minute as Gera squared for Thorne 20 yards out and the 20-year-old flashed a low shot just past the left-hand post as he was charged down by Clint Hill.
Two minutes later Lukaku worked his way to the right byline and cut the ball back for James Morrison, whose goalbound effort was charged down by Ben Haim.
The Baggies were forced into a change on 21 when Gera limped off after a challenge with Ji-Sung Park and was replaced by Fortune.
Morrison was proving a handful in midfield and he worked the ball well with Long in the 27th minute before firing a tame effort straight at Cesar from the edge of the box.
The hosts were struggling to penetrate the Baggies’ backline and just before the half-hour mark Adel Taarabt was reduced to a long-range shot that went high and wide.
Long came close to opening the scoring for the visitors in the 32nd minute as Morrison’s dink eventually fell to him in the box and he forced Cesar into a smart save.
Lukaku picked up the rebound and swivelled before seeing the Hoops goalkeeper safely hold his 15-yard effort low down to his right.
Moments later Long fed the ball inside to Fortune in the area – but the ex-Celtic man scuffed his shot well wide of goal.
Six minutes before the break Fortune could only prod Taarabt’s pass into the path of Jamie Mackie in the box and the Rangers winger saw his shot charged down by Gabriel Tamas.
Seconds later Taarabt pounced on a loose ball and drove forward only to fire a weak 20-yard shot at Myhill with the outside of his foot.
The best chance of the half fell to Lukaku on 41 – but he saw Cesar pull off a fine point-blank save as the on-loan Chelsea striker headed goalwards from Morrison’s cross.
Lukaku returned the favour two minutes before the break as he squared for Morrison and, on the stretch, the Scotland international put the ball inches past the left-hand upright from ten yards.
Stephane Mbia let fly with a 30-yard strike that flew just over the crossbar as the first-half moved towards its climax.
But Clarke was forced into another change in the third minute of stoppage time as Gera’s replacement, Fortune, made way for Jerome Thomas, back from his successful loan spell at Leeds, due to injury.
Harry Redknapp made a substitution at half-time as former Wolves striker Jay Bothroyd came on for Esteban Granero.
Albion started the second half as they finished the first, with Lukaku blazing just wide of the far post from 18 yards within 35 seconds of the restart.
The R’s came close to taking the lead in the 48th minute when Campbell found space in the box from Park’s right-wing cross and headed over.
Morrison showed a superb piece of skill two minutes later to leave Park trailing in his wake before unleashing a dipping 25-yard effort that flew straight into Cesar’s arms.
The hosts looked more threatening but were still struggling to break down an organised Baggies defence.
At the other end it was similar story, with Billy Jones sending a left-foot shot over from 20 yards.
A jinking run by Thomas had the Baggies fans on their feet in the 75th minute – but his shot from the edge of the area lacked the power to beat Cesar.
Albion finally broke the deadlock four minutes later when Thorne’s deflected shot inside the box fell into the path of Long, whose shot from ten yards found the net via a touch from substitute Onuoha.
The visitors appeared to have weathered a late spell of pressure from QPR but Kieron Dyer cruelly levelled in the first minute of stoppage time as he swept home a deep cross at the back post.
QPR (4-5-1): Cesar; Dyer, Nelsen (Onuoha 70), Hill, Ben Haim; Mackie, Park, Granero (Bothroyd HT), Mbia, Taarabt; Campbell. Subs not used: Green (gk), Derry, Ferdinand, Cisse, Faurlin.
ALBION (4-4-2): Myhill; Jones, McAuley, Tamas, Ridgewell; Morrison, Thorne, Brunt, Gera (Fortune 21 (Thomas 45+3)); Long, Lukaku (Dawson 88). Subs not used: Foster (gk), Rosenberg, Jara Reyes, El Ghanassy.
GOALS: QPR – Dyer (90+1); ALBION – Long (79).
BOOKINGS: ALBION – Brunt (foul 87).
REFEREE: Mark Clattenburg (Tyne and Wear).
ATTENDANCE: 8,984 (Albion fans 1,436).
ALBION STARMAN: James Morrison.