Former Spurs midfielder Sherwood 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.
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.
