Gow turned and volleyed home from close range after United failed to deal with Tom Hateley’s 86th-minute corner.
An extra half hour had seemed inevitable after a scrappy second half but Gow produced a rare piece of quality to continue Craig Brown’s impressive reign as Fir Park boss.
The former Falkirk and Rangers striker will need to secure a new contract to ensure he plays in the last four with his deal due to expire in January, but a potential Hampden trip will ensure the finances are available to tie him up.
The quarter-final victory was Motherwell’s second over United in four days after they beat United 2-1 in the Scottish Premier League and both sides made enforced changes with Keith Lasley and Craig Conway failing to recover from injury.
Steven Saunders replaced Lasley while United brought in Scott Robertson.
Peter Houston also brought in Keith Watson at right-back with Sean Dillon moving into central defence to replace the absent Scott Severin.
Both defenders scored own goals at the weekend but Motherwell were fully deserving of their victory as United struggled to cope with the running of their front players.
And, although United created the first chance when David Goodwillie stabbed the ball straight at Darren Randolph following Garry Kenneth’s knockdown, the hosts soon put the pressure on.
Nick Blackman was inches away from converting Jamie Murphy’s driven ball following a typically direct run from Chris Humphrey.
Another low cross from Murphy soon found Gow, but the forward could not get the ball out of his feet and his effort was comfortably held by Dusan Pernis.
Referee Willie Collum, who received threatening phone calls after awarding a dubious penalty to Rangers at Celtic Park on Sunday, faced his first major decision in the 13th minute when Robertson went down inside the box under a challenge from Steve Jennings.
The midfielder claimed for a penalty after being prevented the opportunity to follow up Paul Dixon’s powerful, parried effort, but Jennings appeared to take the ball and Collum made the right decision by waving away the appeals.
United started far better than they had on Saturday with Robertson at the heart of some good possession play and the hosts suffered a blow when Murphy limped off after 20 minutes, with John Sutton coming on.
United continued to edge proceedings and Goodwillie saw a long-range effort held, but Motherwell regained the ascendancy before the break. Gow released Blackman after a sharp one-two but Pernis closed the on-loan Blackburn striker down quickly.
Dixon made way for Barry Douglas and the second half started in scrappy fashion. Motherwell began to take a grip on possession around the hour mark but chances remained scarce.
A Sutton volley flashed wide but the dearth of quality was summed up in a one-minute spell when Douglas hurled a long throw straight behind for a goal kick before the normally accurate Hateley over-hit a free-kick behind Pernis’ goal.
The first save of the second period came in the 77th minute when Blackman worked space 25 yards out before firing a bouncing effort that Pernis pushed wide.
Scotland manager Craig Levein was not waiting for extra-time, leaving with seven minutes left, but Gow made sure everyone else avoided another 30 minutes.
Hateley’s corner bounced off Buaben’s knee and then the front post before coming out to the striker, who turned and volleyed home from close range.
United almost forced a last-minute equaliser when a long throw bounced in front of Goodwillie at the back post but the home defence scrambled clear to secure their first semi-final for five seasons.