Tottenham were guilty of wasting a number of chances in both halves, but the tension inside White Hart Lane lifted with 16 minutes left when Benoit Assou-Ekotto smashed home a deflected half-volley to put the home side ahead.
West Brom, revitalised after the introduction of Romelu Lukaku as a second-half substitute, kept plugging away and the hosts’ defence crumbled in the 91st minute when Morrison swept home to steal a point for the Baggies.
With all the uncertainty surrounding a number of his star players, and last week’s defeat at Newcastle lingering in the back of his mind, Villas-Boas could have done with marking his White Hart Lane bow with a win, but it was not to be.
The London side lacked the fluency displayed during the majority of last season under Harry Redknapp, although there were encouraging performances from Sandro, Gareth Bale and Jan Vertonghen, who thought he had marked his debut by scoring a 94th-minute winner only to see if ruled out for offside.
Villas-Boas and his trio of new signings received a huge round of applause before kick-off, and the Portuguese’s charges looked determined to deliver for their new boss.
Bale was the architect of an early onslaught. The winger, playing in a wider role than he did under Redknapp, cut in off the left flank and fired a shot just wide.
The Welshman then found Sandro in the box but the Brazilian went down under pressure before he could get a shot off.
Bale spun his marker in the corner and sent a low ball across the box but Liam Ridgewell prevented Aaron Lennon steering home from close range.
Lennon did get on the end of a Bale cross soon after but the England winger miscued a volley.
Rafael van der Vaart was the next Spurs player to go close, steering a curled shot just wide after Lennon cut the ball back to the Dutchman.
Tottenham’s dominance on the game started to wane, leaving Villas-Boas to cut the look a worried figure on the edge of his technical area.
West Brom still lacked a cutting edge going forward, however, with Vertonghen snuffing out any threat from the visitors.
The Belgian had a chance to mark his Spurs debut with a goal after playing a clever one-two with Bale but he rifled a volley well over.
Steven Reid fashioned West Brom’s only chance of the first half – a curling free-kick that easily cleared Brad Friedel’s bar, and Ben Foster was called in to action for the first time in the 46th minute when he saved a weak shot from Defoe.
Tottenham continued to dominate in the second period but still could not find the net.
Van der Vaart picked out Bale with a peach of a cross-field pass but the Welshman volleyed well over.
Foster then pulled off a terrific diving save to deny Assou-Ekotto from a 20-yard free-kick.
Defoe then spurned a great chance to put Spurs ahead when he latched on to Jake Livermore’s pass but his effort went wide.
Villas-Boas turned to Emmanuel Adebayor for inspiration, the Togo marksman coming on for Van der Vaart just after the hour.
Villas-Boas thought the £5million striker had won him a penalty when he tussled with Gareth McAuley but Mike Dean waved play on.
Spurs thought they had finally broken the deadlock after 64 minutes when Defoe smashed home an Assou-Ekotto cross, but the linesman ruled the England striker had strayed offside.
Lukaku came on for West Brom and almost made an immediate impact, shrugging off Vertonghen before drawing a good save from Brad Friedel.
Lukaku’s presence was causing Spurs all kinds of problems, but the mood inside White Hart Lane lifted in the 74th minute when the hosts took the lead.
Gylfi Sigurdsson’s corner was cleared in to the path of Assou-Ekotto and he smashed home a half-volley courtesy of a big deflection off McAuley.
Lukaku took advantage of some lackadaisical defending to gain some space just outside the Spurs box but Vertonghen came to the rescue with an outstanding sliding tackle.
West Brom piled on the pressure through Fortune and Lukaku towards the death.
McAuley came agonisingly close to scoring in the final minute when he sent over a looping header that was cleared off the line by Assou-Ekotto.
Then 30 seconds in to injury time West Brom equalised when Morrison swept home from close range shortly after William Gallas had cleared off the line.
Vertonghen thought he had won the game with almost the last kick when he volleyed home but his effort was disallowed.