The Irn-Bru First Division side, who eliminated last season's winners Rangers in the last round, forced a shoot-out through 19-year-old substitute Ally Graham's 118th-minute equaliser in extra-time after it ended 1-1 after 90 minutes.
And after Willo Flood missed his penalty for United, Stewart Murdoch converted for the Bairns to take them through to the semi-finals.
United dominated much of the match but went behind after 70 minutes to Farid El Alagui's effort against the run of play.
Johnny Russell, who looked lively after coming on as a substitute, equalised three minutes later following a strong run down the left and Jon Daly drilled in across goal from Russell's pass early in extra time to put the hosts in front.
United goalkeeper Dusan Pernis saved El Alagui's point-blank effort after 116 minutes but Graham struck two minutes from the end to force a penalty shoot-out and the Bairns, managed by former United defender Steven Pressley, prevailed.
United were seeking to end a four-match winless run dating back to their 2-0 win at Airdrie in the last round, but their poor start to the season continued ahead of Saturday's SPL trip to Dunfermline.
Scott Robertson headed wide after 11 minutes before El Alagui's glancing header was held by Pernis.
United responded and, from Paul Dixon's cross, Dalla Valle's header had Michael McGovern back-pedalling, but the Falkirk goalkeeper tipped the ball onto the bar and out for a corner.
Dalla Valle diverted a Flood drive back into the box wide before Daly latched onto John Rankin's pass, hitting a left-foot volley from the edge of the area off the bar.
United continued to miss chances and on-loan Fulham striker Dalla Valle was replaced by Russell after 56 minutes.
Daly headed over from Flood's corner and Douglas from Keith Watson's cross as United again failed to adequately test Falkirk goalkeeper McGovern.
The Bairns then went ahead against the run of play with substitute Dale Fulton having an immediate impact.
His inswinging cross was met by an El Alagui header which beat Pernis for the striker's 15th goal in 16 games.
However, the lead was shortlived and Russell's persistence down the left side of the box paid off when he drilled a shot low across goal and into the far corner after 73 minutes.
Russell had two further opportunities to find a winner in normal time, but the match went to extra-time.
Robertson was forced off early in the first extra period with Stuart Armstrong coming on, before United went in front after 96 minutes.
Russell this time turned provider, playing the ball through for Daly to shoot in across goal from the right side of the area.
Irishman Daly hit a 25-yard volley from Pernis' long kick onto the bar soon after, with McGovern beaten.
El Alagui shot across the face of goal from a narrow angle before Daly headed just wide from a Flood corner.
McGovern saved a Russell effort from the edge of the area and Garry Kenneth hit a free-kick off target.
Pernis saved El Alagui's point-blank effort four minutes from time following United's failure to deal with a high ball.
Falkirk took the aerial route moments later as McGovern's ball evaded the United defence and Graham, who was introduced after 110 minutes, stole in to prod past Pernis.
McGovern clawed Gavin Gunning's close-range effort round the post moments later as the Bairns took the tie to penalties.
After two successful kicks for each side Flood ballooned his effort over and Murdoch completed a 100% success rate from five efforts as the Bairns progressed.