Thomas Vermaelen struck the decisive spot kick against the post for the Premier League giants, having earlier scored an 87th-minute equalizer to take the game into extra time. The match finished at 1-1 after 120 minutes, despite Arsenal fielding a near full-strength team.
"They have world-class players ... but the way the lads stuck to their job, you have to give them credit," said Bradford manager Gary Parkinson, whose side has won an English-record nine shootouts in a row.
Aston Villa joined Bradford in the semifinals by beating Norwich 4-1.
With Arsenal down in the seventh place in the league and without a trophy since 2005, manager Arsene Wenger's judgment is being questioned like never before in his 16 years at the club and this result will give more ammunition to his critics.
The upset will rank alongside Manchester United's exit at the hands of York in the second round in 1995 and Liverpool's embarrassing defeat in a penalty shootout at home to Huddersfield in the third round in 2010.
Wenger's team failed to produce a shot on target until the 69th minute on a freezing night at Valley Parade, with Bradford having pierced Arsenal's brittle defence through Garry Thompson's fine volley following a 16th-minute free kick.
A late rally saw Vermaelen head home the equalizer from close range, and Santi Cazorla fired a shot against the bar in the second half of extra time that Arsenal dominated.
However, Bradford was never behind in the shootout with Cazorla and Marouane Chamakh denied in Arsenal's first two penalties. Vermaelen still had the chance to take it to sudden death, but his attempt slithered against the goal frame.
"I can't fault the effort from our team," said Wenger, after seeing his side's best route to silverware this season blocked off. "You have to give credit to Bradford. We will get over it — it's part of our job."
Arsenal's owners are unlikely to give up on the well-respected Wenger, who is expected to be given funds to strength his playing staff in the January transfer window.
The lack of a potent striker appears to be the most pressing concern, with Gervinho and substitute Chamakh making precious little impact up front against Bradford. The Gunners were also outmuscled in midfield for the first hour, only gaining the ascendancy in that area once Tomas Rosicky and Alex Oxlade-Chamberlain were thrown on in desperation in the 69th.
Bradford, currently fourth in League Two, became the first fourth-tier team since the 2006-07 season to reach the semifinals of the competition.
Villa scored three times in the final 11 minutes to see off Norwich in a repeat of the 1975 final, with Austrian striker Andreas Weimann netting in the 79th and 85th before Christian Benteke completed the win in stoppage time.
Norwich, on a nine-match unbeaten run in the league, went in front through Steve Morison in the 19th only for Brett Holman to equalize two minutes later.
It sealed a successful return to Carrow Road for Villa manager Paul Lambert, who left Norwich in the summer in acrimonious circumstance and is still in a legal wrangle with his former club.
Swansea plays Middlesbrough on Wednesday, with the other last-eight match coming on Dec. 19 when Chelsea visits Leeds. The semifinals will be played in January.Suggest a correction