The home side set out to give their opponents little space in the opening stages of the game, and nearly caught out goalkeeper Hugo Lloris in the first few minutes when Carlton Cole pressed down on goal.
On the half-hour mark, there was drama as Kyle Naughton was judged to have deliberately struck the ball with his hands in the Spurs box.
The 25-year-old was sent off, and Mark Noble stepped up to take the ensuing penalty, but his effort from 12 yards flew wide.
Nevertheless, West Ham were able to find their way back into the game towards the latter stages, as Ricardo Vaz Te fired just wide following a dangerous cross.
In the 53rd, Lloris had to be at his sharpest, pulling off a double save from the Hammers front line.
The home side found themselves a man down just a few minutes later when James Collins clashed with Emmanuel Adebayor and earned himself his second yellow of the game.
With both sides down to ten men, Christian Eriksen fired a dangerous shot from the resulting set piece, which just swerved over the bar.
In the 70th minute, Andros Townsend struck two quick chances in succession, both of which were thwarted by Adrian.
With Spurs on the ascendency, the Hammers found themselves with their backs to the wall for the remainder of the half, although Noble troubled Lloris after a surge from midfield.
Just as it seemed that the game would end all square, Dier, thrown upfield in an ambitious attempt to cause havoc in the Hammers box, raced onto a through-ball, rounded the keeper and slotted into the unguarded net to break the home crowd's hearts.
