The Heed had lost 3-2 in the first leg but, in front 2,264, fell behind inside the opening five minutes at the International Stadium to make the task even harder.
Scott Barrow's 35-yard volley gave them hope, and Newcastle teenager Lewis McNall came on for the final 21 minutes as Steve Watson's side chased another goal, but Bromley held on to win 4-3 on aggregate and progress to the final at the national stadium.
Callum Williams wasn't involved for the Heed.
In-form Ivan Toney earned the Iron a point against Rochdale. His second-half header was his fourth goal in his last seven games and kept his team in the promotion race - but manager Graham Alexander was surprisingly sacked after the game.
Toney told iFollow Iron: "I have been trying to get on the end of the flick-ons as much as I can. I thought that the game was there for the taking, so I was just trying to score for the team and get us closer to three points. Unfortunately, we didn't get them but we will take a point and move onto the next game."
Meanwhile Alex Gilliead played the full game as Bradford returned to winning ways with a 1-0 success against Gillingham, and Sean Longstaff came on for Blackpool in the 66th minute against MK Dons. He hit the bar in the last minute of a 0-0 draw.
Kyle Cameron completed 90 minutes for Queen of the South as they drew 3-3 with Livingston while Crewe's 2-1 win at Newport County was achieved without Jamie Sterry, who was injured, and Daniel Barlaser, who was an unused substitute.