Spurs right-back Kyle Naughton received his marching orders just before the half-hour mark, when referee Chris Foy adjudged him to have prevented an obvious goalscoring opportunity when he handballed at close range from Kevin Nolan's shot on the turn.
However, Mark Noble missed the resulting spot-kick, sending a low effort wide of Hugo Lloris' right-hand post.
In the second half, Hammers centre-back James Collins was shown a second yellow card and then a red by Foy after blocking the run of Emmanuel Adebayor through the centre of the pitch.
World Cup star Enner Valencia came off the bench for the final 10 minutes for West Ham, after Sam Allardyce's starting strike partnership of Carlton Cole and Ricardo Vaz Te had proved ineffective.
In the closing stages, Nolan cleverly played in Stewart Downing but Spurs goalkeeper Lloris was quickly off his goal-line to make a crucial save.
Then, in the third minute of stoppage time, debutant Dier burst through the Hammers defence onto Harry Kane's through-ball, rounded Adrian and slid home the winner to give Mauricio Pochettino three points in his first competitive game in charge of Tottenham.
Pochettino handed club-record signing Erik Lamela just his fourth Premier League start. The 22-year-old Argentinian failed to make an impact in his first year at White Hart Lane but was given a chance to impress by Pochettino, along with Dier.
There was no place in the starting line-up for Ben Davies, with Danny Rose preferred at left-back for the visitors, who were captained by Younes Kaboul.
Left-back Aaron Cresswell and defensive midfielder Cheikhou Kouyate made their West Ham bows.
During the summer Allardyce had been tasked with bringing a more entertaining brand of football to Upton Park and, up against Pochettino's supposedly free-flowing Spurs side, his team started the brighter.
Midway through the half, Vaz Te miscued when a Collins flick-on found the Portuguese winger in plenty of space at the back post and shortly afterwards, Kouyate's diving header flew inches wide of Lloris' goal.
With just under half an hour on the clock, the hosts were given the perfect opportunity to open the scoring when they were awarded a penalty for handball against Naughton.
The full-back was arguably unlucky to be sent off for the misdemeanour but Noble could only drag the resulting spot-kick wide, with Lloris diving the wrong way.
The Hammers remained the dominant force for the remainder of the half but could not find a way past Lloris, with Vaz Te flashing an effort wide as Allardyce's side continued to struggle in their quest to find the target.
Tottenham came close to breaking the deadlock against the run of play as Lamela stole the ball off Noble and played in Aaron Lennon, who could only whip his shot over the bar.
Vaz Te finally forced Lloris into a save, albeit a routine one, as he headed into the ground and straight into the hands of the France goalkeeper.
After the break, it was Downing who stung the palms of Lloris, with Pochettino deciding to change things on the hour-mark as Andros Townsend and Lewis Holtby were introduced moments before the contest was evened up.
Collins picked up his second yellow card as he stopped Adebayor in his tracks, with referee Foy having a busy first day of the new season.
Christian Eriksen may have fluffed his lines from the free-kick that followed but Spurs were soon in the ascendancy for the first time in the game.
Townsend forced Adrian into the best save of the match before seeing a similar effort deflected off target shortly afterwards.
With both sides a body light, the game opened up as the half wore on - with Noble only able to shoot straight at Lloris after Kaboul had failed to bring the ball out of defence for the visitors.
Allardyce brought on summer signing Valencia for a 10-minute cameo, with the Ecuador striker receiving a rapturous reception from the home fans.
It was Downing, though, who came close to winning it late on for West Ham. The England winger exchanged neat passes with Nolan on the edge of the Spurs box before forcing his way through on goal, with Lloris doing well to close in and block Downing's shot.
Adrian had to be lively at the opposing end to push out a long-range Nabil Bentaleb drive but the Spaniard could do nothing as Dier, a summer signing from Sporting Lisbon, kept his cool to send the travelling Spurs fans into delirium.
Phil Thompson on Tottenham's 1-0 win at West Ham
Being a defender, I wondered what Dier was doing up there in injury time! It’s 10 v 10 and there’s extra work for everybody - and there was the right-back, going right through the middle. I had to have a double take - but it was a fabulous finish for him.
This was a slow burner of a game. I thought the handball for the penalty was too close to be given. James Collins couldn’t have any complaints over two yellow cards. Tottenham didn’t always play like a team last season, but early on everybody was closing down together. That’s one of the things the manager brought to Southampton and you saw the same in this - but there was no cutting edge to the side.