The Bantams knocked out their Premier League opponents on penalties - their ninth shootout triumph on the spin - in Tuesday night's quarter-final at the Coral Windows Stadium.
And former City striker Goodman told Sky Sports that the victory can lift the Yorkshiremen, who tasted top-flight football as little 11 years ago, in their pursuit to get out of the fourth tier.
"I hope this can be a springboard for more important things which is getting back up the Football League ladder," Goodman said of Bradford, who currently sit fourth in League Two.
"They won't care who they play in the semi-finals but they now have a run of winnable fixtures coming up in League Two and can put themselves in a really strong position."
On the game, in which the Bantams took the lead through Garry Thompson after 16 minutes, before being pegged back by Thomas Vermaelen's 88th-minute header, Goodman added:"Bradford deserved it; they were running on empty for the whole of extra time but they hung in there and held their nerve.
"They set their stall out early on and didn't allow Arsenal to dominate them and that was very important, as had Arsenal put their stamp on it in the first 20 minutes and got in front there would only ever have been one winner."
Ex-Bradford winger Peter Beagrie believes City manager Phil Parkinson has made some shrewd signings since he took charge in 2011 - but he admits that Bantams goalkeeper, Matt Duke, was the star of the show against Arsenal.
"Phil has managed to put his own slant on the team," said the Sky Sports pundit. "His acquisitions have been brilliant, taking players who have had good educations at bigger clubs and moulding them, and they have a nice blend of youth and experience."
"Duke - and the whole of the Bradford back four - showed great concentration levels and that's what you have to do as a goalkeeper, not just show your ability and agility.
"Arsenal were going to come on strong as they have quality players, and some of the shots he was saving were not just regulation ones, they were coming through bodies. It was a tremendous display.
"Bradford's supporters have stayed with the club through thin and thinner and I am delighted for them because they are so passionate and turn out in their numbers every single week.
"They've dreamt of moments like this and now they've got a semi-final place, but Bradford have been staring into the financial abyss, too, and hopefully this money will safeguard their future."