Quintanilla built up a one-minute advantage in the early part of the 379km Shubaytah-Haradh test, but was chased down rapidly by KTM rider Matthias Walkner, who assumed a narrow one-second lead at the 324km way point.
But the Chilean rider fought back in the final stretch to the finish line, taking the stage victory by just nine seconds from Walkner.
That allowed him to heavily cut into the advantage of Brabec, who lost nearly 12 minutes after starting second on the road behind Honda teammate Joan Barreda and finished 10th.
Brabec and Quintanilla are now separated by 13m56s with just Friday’s final stage to run.
Behind the leading duo, reigning champion Toby Price (KTM) sits third after finishing fifth on Stage 11, another nine minutes down on Brabec, with Walkner jumping up two places to end up fourth in the overall classification.
Honda duo Barreda and Jose Ignacio Cornejo are placed fifth and sixth respectively ahead of Luciano Benavides (KTM), the latter finishing third in Stage 11.
Franco Ciami is the top Yamaha rider in 8th, while the top 10 is completed by Skyler Howes (Husqvarna) and Andrew Short (Yamaha).
The Dakar Rally concludes with the final stage on Friday, with Honda remaining in a prime position to end KTM’s 18-rally winning streak.
UPDATE: Honda rider Barreda was later handed a 15-minute penalty for an engine change, dropping him from fifth in the overall standings to seventh.
Standings after Stage 11 (Bikes, Top 10):
Jose Ignacio Cornejo
Quads: Sonik quickest, Casale in control
Dakar's 2015 quads champion Rafal Sonik picked up his first stage win in the 2020 edition, outpacing nearest rival Ignacio Casale by three minutes.
Casale's performance was enough to reinforce his lead and he thus heads into the final day with a 21-minute advantage over Simon Vitse.
Sonik, an hour back from Casale, is set to complete the podium and himself has a two-hour buffer to Manuel Andujar.