Dominic Poleon volleyed the winner and his first league goal of the season in the 80th minute after Bournemouth striker Lewis Grabban had cancelled out Ross McCormack's second-half opener.
Leeds had failed to muster a single chance on goal when Bournemouth were harshly reduced to 10 men in the 31st minute.
Goalkeeper Ryan Allsop was adjudged to have brought down Leeds striker Noel Hunt in the penalty area and was shown a straight red card by referee David Coote.
But Bournemouth's replacement keeper Darryl Flahavan saved McCormack's subsequent penalty and misfiring Leeds went on to make hard work of their first win in five matches.
"It reminds me of a game when I was permanent manager at Reading when I first got the job," McDermott said. "We played Barnsley and won 1-0 and I thought then that the only thing that mattered was to win.
"As caretaker we hadn't won for five, we drew two and lost three and we were struggling at the bottom of the table, but we beat Barnsley and went on to win eight on the trot.
"I'd take that again! But we deserved to win. That's never an easy game, they've got nothing to lose, down to 10 men.
"We showed great guts, we were nervous towards the end when they got back to 1-1 - but we didn't go under and created a lot of chances.
"I definitely think we deserved to win. The only thing that mattered in this game was the result. Nothing else."
Allsop's controversial dismissal proved to be the game's turning point, but McDermott insisted referee Coote made the correct decision.
"I've seen it back and it was a penalty," McDermott said. "He (Hunt) was going to score. It was a definite penalty."
The Leeds boss added: "I went through the mill because it was important to get that result.
"We've got it and we'll be better for that on Saturday (against Derby)."
Bournemouth boss Eddie Howe refused to be too critical of referee Coote and was pleased with the response of his 10 men.
Howe, whose side have won only one of their last five league games after Saturday's 3-1 home defeat to Blackburn, said: "I've only seen a replay of the penalty decision once and that was at match speed, so it was a difficult one.
"I feared the worst when I saw the challenge. I think Ryan's got a different version of it, but it was difficult, the referee's had to make a split-second decision.
"I can't see from where I am and he's given the penalty. Unfortunately Ryan's been sent off as well and when you come to a tough ground like this, to go down to 10 men so early in the game, it really was a mountain to climb for us.
"But we scored a great goal through Lewis Grabban and it's a shame we couldn't hold on.
"But there are some real positives from us, the performances of Darryl Flahavan and Andrew Surman, with very little game-time, and Jack Collison on his debut - I thought there were some real positives."