Jimmy Smith gave QPR the lead after he poked home Lee Cook's free-kick.
Luton equalised when Adam Boyd converted Steve Robinson's cut-back before they took the lead when Ahmet Brkovic headed in David Bell's cross.
Markus Heikkinen's own goal from Lee Cook's corner made it 2-2 before Blackstock turned in Marc Nygaard's flick on from Smith's corner.
Luton manager Mike Newell on his relationship with chairman Bill Tomlins:
""I have not spoken to the chairman for months. I cannot understand what he is doing here. "The club has gone backwards off the pitch during the last two years. He has not made any decisions, and he has contributed nothing.
"Everything he has got involved in, he has messed up. I am not considering my own future, because, apparently, he is going. I will not suffer fools - I never have, and I never will.
"There are players out of contract at the end of the season, but nothing is being done. I am up to my eyes in it."
QPR defender Marcus Bignot:
"Comments like those will stop female officials from trying to make their way in the game. "Amy Rayner is a role model, and full credit goes to her. What would he have said if it had been a male assistant referee making that decision?
"It's easy to blame officials, and Mike Newell is entitled to his opinion, but remarks like that do not help the game."
Luton: Brill, Foley, Barnett, Heikkinen, Emanuel, Edwards, Robinson (Morgan 85), Bell, Brkovic, Vine (Feeney 85), Boyd.
Subs Not Used: Barrett, Perrett, Holmes.
Goals: Brkovic 45, Boyd 45.
QPR: Royce, Mancienne, Rehman, Stewart, Bignot, Ainsworth, Bailey, Smith, Cook, Blackstock (Gallen 80), Nygaard, Gallen (Ward 89).
Subs Not Used: Paul Jones, Milanese, Baidoo.
Booked: Blackstock.
Goals: Smith 33, Heikkinen 51 og, Blackstock 54.
Att: 9,007
Ref: A D'Urso (Essex).
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