Lukas Jutkiewicz missed an early Blues penalty after Che Adams was fouled.
Adams was then sent off following a 23rd-minute challenge on Martin Cranie but Jonathan Grounds put Birmingham ahead four minutes before the break.
Craig Gardner doubled the lead from Blues' second penalty, following Dean Whitehead's foul on Jacques Maghoma.
Victory kept Birmingham out of the relegation zone and, with just one match remaining, they are two points above Blackburn and Nottingham Forest but have an inferior goal difference.
In a tense atmosphere in front of a 26,914 near full house, Blues wasted a glorious chance with their missed eighth-minute penalty, after Adams was brought down by Mark Hudson.
Joel Coleman dived to his right to save Jutkiewicz's low spotkick - and worse followed when Adams was given his straight red card by referee Mike Jones.
But, against the odds, despite being a man short and having lost club captain Paul Robinson for his indiscretion in last weekend's 1-0 local derby defeat at Aston Villa, the hosts rallied.
They took the lead when Gardner's right-wing cross was headed back across the goal by Jutkiewicz and Grounds forced the ball home at the near post - for only his second goal of the season.
Then boyhood Blues fan Gardner scored only his second goal since returning to the club in January when he took over penalty-taking duties after Maghoma had been brought down.
Huddersfield, who made 10 changes after being assured a play-off place with Tuesday's 1-0 win at Wolves, drop two places to fifth ahead of next Sunday's final home game against ex-Terriers boss Neil Warnock's Cardiff.
This was only City's third win in 24 Championship games, lifting them a place to 20th as Forest lost at QPR.
Blues have a considerably worse goal difference than both Forest and Blackburn going into the final day on Sunday, 7 May.
New manager Redknapp and his assistant Steve Cotterill must now take their side to Bristol City, where Cotterill was manager from December 2013 to January 2016, knowing only a win will guarantee safety.
If either 21st-placed Forest or 22nd-placed Blackburn fail to win then Blues will be safe regardless of their own result at Ashton Gate.
But Forest, who are at home to Ipswich, and Rovers, who are at Brentford, will send Redknapp's side down if they both earn final-day victories while Blues fail to win.
Birmingham caretaker boss Harry Redknapp told BBC WM:
"Amazing, To miss a penalty in the first 10 minutes, then have a man harshly sent off, it was an awful start, but the lads showed a lot of character.
"We defended fantastically. The back four stood strong, as did our four across the middle. They did their jobs and Lukas Jutkiewicz up front held the ball up well and caused problems when he had the chance.
"It was a great team effort and Michael Morrison gave a real captain's performance. He put himself up to play and proved a real leader for us."
Huddersfield boss David Wagner:
"At Wolves I made five changes and everyone knows my rotation policy is part of my idea in a busy Championship.
"Changes were necessary with our thoughts to the play-offs as no one knows what will happen over the next month.
"It is unfair talking about my starting XI because I wanted fresh players.
"At the moment we are not getting consistent results but we have been able to stay in the top six and progress to the play-offs for the first time in 45 years."
Match ends, Birmingham City 2, Huddersfield Town 0.
Second Half ends, Birmingham City 2, Huddersfield Town 0.
Attempt saved. Jack Payne (Huddersfield Town) left footed shot from outside the box is saved in the centre of the goal. Assisted by Tommy Smith.
Substitution, Birmingham City. Josh Cogley replaces Lukas Jutkiewicz.
David Davis (Birmingham City) is shown the yellow card.
Nahki Wells (Huddersfield Town) wins a free kick in the attacking half.
Foul by Michael Morrison (Birmingham City).
Corner,  Huddersfield Town. Conceded by Jonathan Grounds.
Foul by Tommy Smith (Huddersfield Town).
Lukas Jutkiewicz (Birmingham City) wins a free kick in the attacking half.
Offside, Huddersfield Town. Nahki Wells tries a through ball, but Dean Whitehead is caught offside.
Delay over. They are ready to continue.
Delay in match Philip Billing (Huddersfield Town) because of an injury.
Attempt missed. Tommy Smith (Huddersfield Town) left footed shot from outside the box is too high. Assisted by Collin Quaner following a corner.
Corner,  Huddersfield Town. Conceded by Jacques Maghoma.
Attempt blocked. Sean Scannell (Huddersfield Town) right footed shot from the right side of the box is blocked. Assisted by Jack Payne.
Attempt blocked. Jack Payne (Huddersfield Town) left footed shot from outside the box is blocked. Assisted by Collin Quaner.
Goal!  Birmingham City 2, Huddersfield Town 0. Craig Gardner (Birmingham City) converts the penalty with a right footed shot to the bottom left corner.
Penalty conceded by Dean Whitehead (Huddersfield Town) after a foul in the penalty area.
Penalty Birmingham City. Jacques Maghoma draws a foul in the penalty area.
Foul by Tommy Smith (Huddersfield Town).
Lukas Jutkiewicz (Birmingham City) wins a free kick in the attacking half.
Foul by Sean Scannell (Huddersfield Town).
Jonathan Grounds (Birmingham City) wins a free kick in the defensive half.
Sean Scannell (Huddersfield Town) wins a free kick on the right wing.
Foul by David Davis (Birmingham City).
Substitution, Huddersfield Town. Aaron Mooy replaces Mark Hudson.
Substitution, Huddersfield Town. Sean Scannell replaces Harry Bunn.
Offside, Birmingham City. Craig Gardner tries a through ball, but Jacques Maghoma is caught offside.
Foul by Mark Hudson (Huddersfield Town).
Lukas Jutkiewicz (Birmingham City) wins a free kick in the attacking half.
Attempt missed. Nahki Wells (Huddersfield Town) right footed shot from the centre of the box is high and wide to the right.
Jonathan Grounds (Birmingham City) is shown the yellow card.
Hand ball by Dean Whitehead (Huddersfield Town).
Philip Billing (Huddersfield Town) wins a free kick in the defensive half.
Foul by Lukas Jutkiewicz (Birmingham City).
Attempt blocked. Jacques Maghoma (Birmingham City) left footed shot from the centre of the box is blocked. Assisted by David Davis.
Attempt saved. Lukas Jutkiewicz (Birmingham City) right footed shot from the right side of the box is saved in the bottom left corner. Assisted by David Davis.
Collin Quaner (Huddersfield Town) is shown the yellow card for a bad foul.
Foul by Collin Quaner (Huddersfield Town).
Struggling Birmingham City earned their first win under Harry Redknapp with 10 men as they beat Championship play-off favourites Huddersfield Town.