Thankfully for the pair, Ben Purrington struck before half-time to tie things up to set up a nervous final 45 minutes in Wembley.
However, with 93 minutes on the clock, Charlton fashioned one last opportunity and as Sunderland failed to clear a high Josh Cullen ball into the box, Bauer pounced.
Sunderland scored in the first half in 26 of their 46 games this season, with only League One champions Luton Town managing more goals before the break than the north-east outfit.
But Sunderland have fallen short, falling at the final hurdle, the play-off final in front of 76,000 supporters.
The draw, which would trigger extra time, is 43/20 (3.15).
Monday sees the Championship play-off to decide whether Derby County or Aston Villa will be the 20th Premier league club next season buyt of course there is "local" interest in today's match. "I was a Sunderland boy, living the dream playing for my local team, and I just didn't want to be the person responsible for us losing such an important match".
It was a heartbreaking and cruel way for Sunderland to lose but ultimately they hadn't done enough over the course of the 90 minutes to win promotion.
Cullen took a shot free-kick from substitute Jonny Williams and drove upfield, exchanging passes with Joe Aribo before crossing for Bauer, who stabbed home after his initial header was blocked. I'm devastated for them.
However, they were buoyed by a trip to Wembley for the Checkatrade Trophy final and their recent familiarity with the stadium could prove a factor when they return on Sunday.
"To bring all this back together, it has been some season".
"Right now, that feeling of disappointment is the over-riding one, because of the manner we've lost the game".
"We've had that terrible experience twice in two months, in very hard circumstances".