Spurs were facing a 10-point deficit to Leicester City as they waited for a delayed kick-off at White Hart Lane after the leaders beat Sunderland 2-0 earlier on Sunday.
The late arrival of the Manchester United team bus was followed by an uneventful first-half but Dele Alli's opening goal with 20 minutes to play saw the contest explode into life.
Toby Alderweireld powered home a header from Erik Lamela's free-kick before the Argentina winger completed a chastening six minutes for Louis van Gaal's beleaguered side by sweeping in the third.
"We'll keep on fighting until the end in every single game. That's what we expect of ourselves," Dier told Sky Sports.
"No matter what happens with everyone else we're going to concentrate on ourselves and try and get five wins from now until the end of the season.
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"[Leicester winning] focused us. There's a lot of ground to make up – if they lost or won we needed to win today. We needed to be at our best and we were in the second half.
"When we had the ball they made it very difficult for us in the beginning but we managed to sort that out.
"Credit to Leicester, they don't seem to slip up, but we'll keep doing our best and see where we end up."
Alli continued his superb breakthrough season by scoring an eighth Premier League goal this term, the day before his 20th birthday.
The England international told Sky: "It's important that we keep winning games and finish the season strongly on a good run.
"They were never going to make it easy for us but we knew the chances would come if we were patient.
"My target was to get to 10 this season but I'll take eight."
Alli added: "It's important we don't get too focused on the league. We've had a great season together as a team and whatever happens it's been a great year for us."