Adam Barrett dedicated victory over 10-man Leeds to recently-departed manager Neil Harris after Millwall began life in new hands with a 2-1 victory.
Jed Wallace's early penalty put the hosts ahead, with the incident seeing Gaetano Berardi dismissed for Leeds with just 14 minutes on the clock.
Harris announced his resignation after more than four years in charge on Thursday.
"It's been a whirlwind 24/48 hours and we wanted to put a performance on for Neil and [former assistant] David Livermore, who've been instrumental and turned the club around with an infrastructure in place," said first-team coach Barrett.
"Neil and David brought me in here. I'm forever thankful for that. To get a victory for him meant an awful lot.
"Leeds were second in the table so not the ideal team to play. We had a perfect atmosphere, it was a quick turnaround from Luton in midweek but the character shown was amazing.
"It was a good game because it was Leeds and extra added spice.
"I think the penalty was fair, the referee has given it and we've got on with it. The first half was excellent nonetheless.
"It was about trying to calm everyone down at the break. Everything we said didn't happen in the second half. Leeds scored very quickly.
"We had to regroup and I'm proud of the boys. Leeds took the handbrake off but we limited them to hardly any attempts.
"We grafted and came strong at the end again. With composure we might have killed the game off."
The post-Harris era began perfectly when Bradshaw got the wrong side of Berardi, with the defender seeing red after bringing him down.
Wallace smashed home the resulting penalty for his third goal in five games against Leeds, with Bradshaw doubling the lead when applying a deft touch into the corner.
Alioski made it 2-1 just moments after the restart but Leeds couldn't grab an equaliser, despite Mateusz Klich's curled effort.
"The result should be different. Each time we lose always I find few logical reasons to justify it," said Leeds head coach Marcelo Bielsa.
"For me, it's always a shame to explain why we lost. I think with 10 men we were better than them.
"Always, we are trying to explain why what we want doesn't happen. It impacts the tolerance of the people who are listening.
"There were three important situations: the penalty, the red card and the next situation where Jack Harrison went down in the box.
"I prefer to analyse the game and not the referee. The second goal for them was a big impact on the match.
"After the first half, I decided to improve the structure and organisation in the team, but I should have taken the decision in the first half.
"Even if I didn't take those decisions in the middle of the first half, we were better with one less player, but maybe we realise more this superiority in the second rather than the first half. Maybe I could have taken this decision earlier to help the team.
"It's true we created few goal chances, but it's also true we had some situations that were dangerous, even if no chances were created."