Adebayor scored two crucial goals in last week's 3-2 victory over Southampton, but he proved to be the foe on a rather underwhelming introduction to full-time management for Sherwood.
Less than two minutes after Christian Eriksen put Spurs ahead, Adebayor woefully miscued a Morgan Amalfitano cross and Jonas Olsson tapped in easily from close range to secure a valuable away point for the Baggies.
Sherwood's appointment, on an 18-month contract, raised a few eyebrows when it was announced on Monday.
The former Spurs midfielder made his desire to bring attacking football back to White Hart Lane clear in his programme notes.
The home fans applauded the team's attempts to get the ball forward, but from this performance, it is clear that much work still has to be done if Sherwood is to achieve his goal of leading Tottenham into the Champions League.
Adebayor had an off day up front while Roberto Soldado also failed to deliver. Eriksen's movement, and his stunning free-kick, deserved praise.
But an attacking intent left gaping holes in the midfield which Albion exploited.
Had Matej Vydra had a better game, or Hugo Lloris a worse one, this could have been another damaging defeat.
The smattering of boos that rang out at the final whistle showed that Spurs fans have their concerns too.
Tottenham go into the final match of the 2013 three points behind fifth. The gap to the top four will probably widen after tonight's game between Manchester City and Liverpool.
West Brom will take confidence from this result, though, and whoever takes over as their new manager - with Pepe Mel the favourite - will be confident of avoiding the drop.
At the start of the afternoon 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.
It took him just two minutes to get to his feet, though, as Spurs made a slow start.
The Tottenham manager shouted at his players, telling them to get the ball out wide and penetrate the Albion defence.
Sherwood lost his patience, yelling an expletive as West Brom took advantage of a wide-open midfield to run at the Tottenham back line.
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 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 Soldado's goal-bound shot.
Tottenham's fans then got the goal they yearned just after the half-hour.
Referee Anthony Taylor blew for a late tackle by James Morrison on Kyle 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 Steven Reid and Gera missed another chance from six yards.
Tottenham resorted to long shots from Danny 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.
Erik 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.