The hosts went ahead through a fine free kick by their Danish summer signing, but the Baggies' Swedish defender hit back immediately to deny Spurs' new boss a second straight win
The former Spurs midfielder was handed the task of replacing Andre Villas-Boas after Sunday's 3-2 win at Southampton, but they found scoring much tougher in front of the White Hart Lane faithful.
Christian Eriksen had put them ahead with a thunderous 20-yard free-kick - his first league goal since moving to Tottenham - but Jonas Olsson bundled home a deserved equaliser almost immediately.
West Brom's interim coach Keith Downing strengthened any hopes he has of becoming Steve Clarke's full-time successor after a gritty display, although the Midlands club are now eight games without a win.
Spurs racked up 20 shots to their opponents' total of six, but the profligacy which has plagued their season again reared its head.
The hosts came into the game unbeaten in their last eight Boxing Day fixtures and made two changes - Nacer Chadli and Lewis Holtby replacing Mousa Dembele and Erik Lamela in an attacking set-up.
West Brom handed a full Premier League debut to Matej Vydra after his late heroics against Hull City last time out, in one of seven changes as they switched to a 3-5-2 formation.
The visitors looked for counter-attacks as Spurs dominated possession and Vydra was twice denied by Hugo Lloris as he sprung through a creaky looking home defence.
Spurs were threatening as they surged forward with numbers, Holtby and Gylfi Sigurdsson lashing long-rangers wide, while Roberto Soldado saw a dangerous effort blocked away.
Vydra wasted a great chance on the half-hour as he nodded wide and was made to pay six minutes later as Eriksen sumptuously opened his Premier League account from the edge of the area.
Tottenham's lead lasted less than two minutes, though, Olsson turning home after the hosts failed to clear Morgan Amalfitano's free-kick.
Danny Rose thundered a 30-yard effort over the crossbar six minutes after the half-time break, while Lloris had to be alert to fist away Steven Reid's fierce free-kick with both sides looking for a second goal.
Lewis Holtby dragged a shot wide after the hour mark and was replaced by Nabil Bentaleb moments later, with the young Frenchman's presence giving Spurs a platform from which to dominate the closing stages.
Ben Foster displayed fine agility to keep Kyle Walker's long-range thunderbolt out and Sigurdsson could only nod substitute Lamela's cross into the side-netting as the win looked to be slipping out of Spurs' grasp.
Soldado demanded a penalty after his cut-back was blocked by a sliding Olsson's arm, but the Spaniard was booked himself for a scything tackle on the Swede after Anthony Taylor waved away his appeals.
It was ultimately the last action of the match as Tottenham slipped to eighth in the Premier League table.