Jermain Defoe's 100th goal for Tottenham looked to have given the home side the win after Roman Pavlyuchenko had equalised Peter Odemwingie's fifth-minute opener.
But with nine minutes remaining, Cox dipped a wonderful 20-yard shot over Heurelho Gomes' head to steal a point and leave Spurs one point behind fourth-placed Manchester City, who play their match in hand against Blackburn on Monday.
The result was a bitter blow for Harry Redknapp and his men, who had their tails up after Wednesday night's thrilling comeback against Arsenal.
Spurs will now hope that relegation-threatened Rovers will surprise Roberto Mancini's men and it may be that the Londoners have to win all five of their remaining matches - which include tough fixtures against City, Chelsea and Liverpool - to make it into the Champions League for the second year running.
Gareth Bale handed Redknapp a boost before the kick-off when he overcame a knee injury to make a starting XI that also included Younes Kaboul and Defoe, who had moved onto 99 goals on March 6.
Wednesday's north London derby may have been a thrilling encounter, but the two points dropped gave Saturday's clash added importance, especially given that City were not in action until Monday.
Theo Walcott's strike had put Spurs behind five minutes into Wednesday's 3-3 draw and it was the same scenario again as Odemwingie latched on to Carlos Vela's pass to outmuscle Benoit Assou-Ekotto and curl a delightful 20-yard shot past Gomes for his 13th of the season.
Assou-Ekotto picked up what looked to be a muscle strain in the challenge and was replaced by Sandro as Bale dropped to left-back.
Redknapp urged his side to find an instant reply but the hosts' play bore no hallmarks of the high-tempo style they deployed to such good effect in Wednesday's comeback.
Inconsistency in front of goal was also a problem. William Gallas spooned a header over from five yards before Pavlyuchenko miscued a diving header from the same distance moments later.
Redknapp cut a frustrated figure on the touchline but he was punching the air after 26 minutes when Pavlyuchenko equalised.
The Russian stole the ball off a West Brom defender and rifled past Scott Carson with a 20-yard strike that went in off a post.
Spurs then upped the pressure on their opponents as they looked to take the lead. First Bale curled a 25-yard free-kick just over before van der Vaart stung Carson's palms with a fierce drive. The rebound fell to Pavlyuchenko, but he blazed over from 18 yards while the Baggies goalkeeper was struggling to get to his feet.
Sandro was putting in a steely performance since his introduction, breaking up West Brom's play as the hosts looked for a second.
Defoe was struggling to get into the game but skipped past two Baggies defenders before being shrugged off the ball in the box.
Five minutes into the second half Nicky Shorey found Odemwingie unmarked in the box but Gomes pulled off a top-class save to tip his header around the post.
The home crowd demanded the introduction of Aaron Lennon and Redknapp heeded their advice and brought on the England winger for Huddlestone just after the hour.
Luka Modric slipped in the box as he wriggled past his marker and Tottenham did not have the same fluency that they displayed towards the end of the first period.
Lennon won a free-kick on the edge of the box but Van der Vaart's 25-yard effort sailed over.
The home side's frustration at not finding a second ended after 66 minutes when West Brom paid the price for backing off Defoe, who found the bottom right-hand corner of Carson's goal with a well-placed strike.
Every one of Defoe's team-mates aside from Gomes ran across the pitch to congratulate the striker on reaching his ton for the club. The strike was also his 100th Premier League goal and was the club's 1000th in the division.
The ball appeared to strike Michael Dawson's arm as he blocked James Morrison's shot in the box in the 73rd minute but Stuart Attwell did not give a penalty.
Defoe breezed past three Baggies defenders before being shrugged off the ball in the box. Christ Brunt then picked the ball up at the other end before smashing a 35-yard drive inches wide, with Gomes well-beaten.
West Brom had the shock equaliser nine minutes from the end when Cox curled a shot that flew over Gomes' head to make it 2-2 with his first strike since October.
Redknapp replaced Pavlyuchenko with Crouch with three minutes left as Spurs desperately sought the winner they needed to leapfrog City.
Dawson's frustration boiled over just a minute later when he was booked for a rash challenge on Odemwingie and the home side's push for a winner petered out, much to Redknapp's frustration.