The point lifted Leeds above West Ham into 15th place, but they sit just one point above the relegation zone, while Brentford extended their club record unbeaten league run to eight matches.
Jesse Marsch's side were in dire need of all three points after winning only two of their previous 15 in the top flight.
Brentford's David Raya was the busier goalkeeper - he produced the game's decisive save when turning aside Willy Gnonto's second-half effort - but Leeds lacked the quality in and around the penalty area to seriously test the visitors.
Leeds club record signing Georginio Rutter remained on the bench and that only added to the home fans' irritation.
Brentford, who headed into this weekend's round of fixtures above Liverpool and Chelsea in the table, climbed back into eighth place.
A cagey first half played out between two teams of contrasting styles made for a disjointed affair of few clear-cut chances.
While January signing Max Wober appeared to add much-needed stability for Leeds in central defence on his first Premier League start, neither side was seriously tested.
The action at Elland Road these days tends to lean towards pulsating and chaotic, but the first 45 minutes was anything but, with no shots on target at either end of note.
The highly-organised visitors were happy to dig in and look to hit Leeds on the counter, while the hosts lacked any potency on the edge of the area.
Leeds stepped it up at the start of the second half. Rodrigo's dangerous low cross was hacked clear after good work by Luke Ayling and Brenden Aaronson's curling shot was held by Brentford goalkeeper Raya.
Rodrigo then forced a diving save from Raya and as the Elland Road faithful responded as Leeds began to get on top.
Gnonto's angled effort was well saved by Raya, who then held Ayling's cross and Jack Harrison's drive.
Patrick Bamford replaced Rodrigo with 11 minutes left and played in fellow substitute Luis Sinisterra, but the latter was crowded out as the ball got stuck under his feet.
Leeds fans then held their breath as Brentford right-back Mads Roerslev stole in at the far post in the closing stages only to fire into the side-netting.