The former Sheffield Wednesday player cancelled out Luka Modric's opener four minutes before the interval.
Harry Redknapp's side dominated the first half with Gareth Bale causing constant problems down the left flank.
But they lacked the killer instinct in front of goal, and appeared to miss the influence of Jermain Defoe, who will be out for three months with the ankle injury suffered on England duty.
It was a different story after the break with the Baggies more than matching their north London opponents and they came close on three occasions in the final 10 minutes to snatching a winner.
Tottenham started brightly and Bale got to the by-line and then turned back inside before his cross picked out Modric. His first-time flick flew high over Scott Carson's crossbar but it was an early warning to the home side.
Aaron Lennon should have done better after seven minutes when a back flick from debutant Rafael van der Vaart found him in space 12 yards out and he sent his shot wide.
Bale was troubling the Albion defence and one superb cross just failed to pick out the run of Lennon.
It came as no surprise when Modric put Spurs ahead after 27 minutes. Van der Vaart's lofted cross hit the back of the head of Gabriel Tamas under pressure from Roman Pavlyuchenko.
Modric was first to react to the the loose ball and beat Carson from 10 yards out with a low drive into the corner of the net.
But it was his last meaningful contribution as he was replaced five minutes later by Niko Kranjcar after suffering a leg injury in a challenge with Brunt.
Tamas had a chance to make amends for his error but he headed just wide from Brunt's corner.
Carson held onto a dipping free-kick from Kranjcar as Spurs continued to dictate play and the home side were chasing shadows.
It was completely against the run of play when Brunt equalised after 41 minutes.
Spurs goalkeeper Carlo Cudicini beat out a shot from Marc Antoine-Fortune but the ball looped up in the air and Peter Odemwingie got a touch before Brunt headed home from close range.
Cudicini then got his body behind a well struck drive from Odemwingie.
Gonzalo Jara was booked for a late challenge on Van der Vaart just before the interval.
Tottenham began the second half on the offensive and Carson had to make two saves within five minutes of the restart.
The England keeper did well to parry an attempt from the ever-dangerous Bale after he had surged into the box from the left flank.
Then Carson was forced to block a powerful 35-yard free-kick from Younes Kaboul.
But Albion tried to retaliate and William Gallas was yellow carded for pulling back Odemwingie after 54 minutes.
The home side were now more in the game than in the opening 45 minutes although Brunt was yellow carded after an hour for deliberate hand ball.
Gallas fired over with a shot on the turn before Robbie Keane replaced Pavlyuchenko.
Chances were now few and far between although Tom Huddlestone dragged two attempts wide from the edge of the box.
The Spurs skipper came to the rescue when blocking a volley from Jerome Thomas with four minutes remaining.
Albion came close to snatching all three points with Cudicini blocking Jonas Olsson's close-range header and turning aside a curling shot from Odemwingie.
Then in the dying seconds Tamas shot wide from 12 yards out.