The hosts made a strong start to the Boxing Day encounter, but the opening goal came from a set-piece on 36 minutes.
James Morrison was penalised for a foul on Kyle Walker – and Eriksen took full advantage by curling a 25-yard free-kick over the Albion wall and in off the underside of the crossbar, giving goalkeeper Ben Foster no chance at all.
The lead only lasted two minutes, though, as Spurs failed to clear a free-kick into the box from Morgan Amalfitano, with the ball making its way through to Olsson who stabbed home past Hugo Lloris from close range.
Morrison spurned a great chance to make it 2-1 just before the break when he dragged a right-footed shot wide of the upright.
But Albion were lucky to have 10 men on the pitch for the last 23 minutes as Steven Reid, who was already on a yellow card, mistimed a tackle on the impressive Danny Rose, but got away with a warning from Anthony Taylor.
Spurs pushed hard for a winner in the closing stages, sending on Erik Lamela, but their final ball was poor and the Baggies banked a deserved point.
At the start of the afternoon, new Spurs boss Tim Sherwood cut a calm figure as he emerged from the tunnel to take the seat he is contracted to occupy for the next 18 months.
But within two minutes he was fuming as Albion took advantage of a wide-open midfield to run at the Tottenham back line. Matej Vydra and Zoltan Gera played a one-two on the edge of the box, but Lloris raced off his line to tip the former Watford man’s shot wide.
Sherwood’s nerves frayed further when Vlad Chiriches’ mistake gave Gera a run at goal but Michael Dawson came to the rescue.
Vydra found a way through the Spurs defence again but Lloris again raced off his line to deny the striker.
Tottenham’s supporters were getting anxious. The attacking intent was easy on the eye, but there was no end product and the open midfield made the home defence look hugely vulnerable.
Nacer Chadli raised spirits momentarily when he sent a dipping ball into the box, but Craig Dawson cleared before Emmanuel Adebayor could convert.
Gylfi Sigurdsson had a powerful shot at goal, but Olsson’s face got in the way. The Baggies captain felt the full force of the shot and needed a couple of minutes to come around.
When play resumed, West Brom went on the attack again and Vydra and Diego Lugano missed Liam Ridgewell’s cross by a matter of inches.
The former Birmingham man fulfilled his duties at the other end, throwing his body on the line to stop Roberto Soldado’s goal-bound shot.
Tottenham’s fans then got the goal they yearned just after the half-hour. Referee Taylor blew for a late tackle by Morrison on Walker. Eriksen whipped the ball over the wall and it hit the crossbar, bounced up off the ground and hit it again before nestling in the net.
Albion were incensed with the original decision against Morrison, but were level less than two minutes later. Eriksen was penalised for a foul on Morrison. Amalfitano swung over a free-kick, Adebayor swung at the ball but failed to clear and Olsson had the easiest of tap-ins.
After the break West Brom continued to threaten. Lloris saved acrobatically to deny Reid and Gera missed another chance from six yards.
Tottenham resorted to long shots from Rose and Walker. Sherwood replaced Holtby with 19-year-old midfielder Nabil Bentaleb.
Reid was lucky to escape a second booking for a late sliding challenge that left Rose rolling around on the floor in agony.
Lamela came off the bench to try to turn the game in Tottenham’s favour but still the hosts were weak in front of goal.
Soldado went close inside the box on two occasions in the dying minutes, but there was no way through and a few boos rang out at the end.