Boro took the lead on 21 minutes when Adama Traore cut in from the left and rode two challenges before planting an unstoppable low drive just inside the near post.
But the Bees stormed back into it just after the half-hour when midfielder Lewis Macleod beat Darren Randolph in the visitors' goal with an almost carbon-copy strike from the edge of the box.
Brentford had to see out the final four minutes of injury time a man down after 75th-minute substitute Romaine Sawyers picked up two yellow cards in the space of five minutes.
Neal Maupay might have given the hosts the perfect start 10 minutes in but his first-time effort was high and wide.
Jonny Howson pounced on a poor John Egan clearance after 20 minutes but saw his angled shot from five yards blocked at point-blank range by Bees keeper Daniel Bentley.
Traore's pace and power was always a threat to the Londoners and Ryan Woods picked up an early yellow for dragging him back before he opened the scoring in style.
The dangerous Spaniard came close to making it two minutes after his opener with a cheeky near-post effort, but Bentley was alert and saved well at the foot of his post.
Macleod, making his first start after a lengthy injury, was pulling the strings for Dean Smith's side and he stung Randolph's palms with a rising drive on the half-hour mark - a warning shot ahead of his excellent equaliser four minutes later.
Maupay might have done better after he latched on to a well-worked one-two on the edge of the box, but his instinctive effort was easily saved by the Boro stopper.
After the break, Brentford came out firing and pinned Tony Pulis' side back for long spells as they nullified Traore's threat in the swirling snow.
Maupay fired just over after a neat build-up down the right and Traore similarly blazed over at the other end.
Macleod's 68th-minute curled effort lacked the power to trouble Randolph and an almost identical shot from substitute Sergi Canos a minute later drifted just wide.
Sawyers' first involvement when he came on in the 75th minute was almost to score, but his powerful grasscutter from 18 yards was superbly tipped wide at full stretch by Randolph.
Brentford pressed higher and higher as they went in search of their first league win over Boro since 1938 but lacked the cutting edge to break down a resolute Boro defence.
Sawyers was booked in the 87th minute and again for blocking Traore's run as the game entered stoppage time.
The visitors - now six games unbeaten - almost nicked the points deep into injury time but Bentley back-pedalled to tip Grant Leadbitter's impudent chip from the edge of the box over the bar.
“I thought they were the better team but we’ve worked really hard defensively and did a smashing job in restricting them to shots from outside the box,” Boro boss Tony Pulis said.
“They are a decent side and Dean Smith has done a smashing job here. He keeps producing good players, sells them and producing new ones.
“I’m disappointed in some ways because we can play much better than that, but this is an old-fashioned ground with fans tight to a pitch which looks small... but they made it look very big for us.”