Colton Fretter netted the game-winning goal with just 34 seconds left to deny Blaze a hard-earned point with overtime looming.
The game was delayed by more than an hour due to the late arrival of Blaze, who were stuck in traffic following an accident on the M1.
It meant a 10-minute warm-up for Blaze, who welcomed Boris Valabik back into the line-up alongside David Clements. Steven Chalmers missed out through injury.
Fretter and Mathieu Roy twice put Steelers in front on the powerplay, but Drew Fisher (powerplay) and Carl Lauzon levelled.
Just when it seemed Blaze would take the game to overtime, Fretter took advantage of a fortuitous deflection to pounce for the winner.
Weber said: “I could not be more proud of this team. Next to the Tychy game this is the proudest I’ve been of this group.
“We could very easily have gone out and said ‘it’s not our day’ and folded up the tent, especially when they scored those two powerplay goals early.
“I loved the spirit of our team. We battled, we talked, we played hard, we blocked shots and we had a ton of scoring chances.
“It took us five hours to get here, it was a crazy trip. But I told the boys to control what’s in our control and that was our preparation, our attitude and our work ethic – all three were fantastic.
“It was heart-breaking for the boys but I won’t let that get us down. There were a lot of positives to take moving forward.”
Blaze next face Cardiff Devils at the Skydome on Sunday night.
Weber added: “On the road against a quality hockey club, I told the boys last year in this building we lost a game with two seconds left and after that we really picked up our game.
“We will take those lessons from our past and get prepared for Sunday. I am not going to dwell on it.
“Cardiff come out fast. They roll their three lines, all three are dangerous. They have one of the best sets of defencemen, they have a goalie who has a saves percentage in the top three and they are a hot team. It is going to take a complete team effort to get points in our building.
“But if this group continues to play with this spirit we are going to win more games than we lose.”