Aston Villa's Jordan Ayew opened the scoring on 62 minutes, curling the ball past Ley Matampi in the Leopards goal.
Paul-Jose M'Poku equalised for DR Congo with a sensational long-range effort from outside the box.
West Ham's Andre Ayew then coolly dispatched a winning penalty, sending Matampi the wrong way to seal the win.
It marks Ghana's sixth consecutive Nations Cup semi-finals appearance, as they seek to end an agonising 35-year wait for a continental title.
In a game that only properly sparked to life in the second half in Oyem, DR Congo - and their captain Dieumerci Mbokani in particular - were left to rue several missed chances early on.
The Hull City striker's best opportunity came on six minutes, after a terrible mix-up in the Ghana defence, but he hit the post when he had the goal at his mercy.
Mbokani then teed up the competition's top scorer Junior Kabananga on 25 minutes, who fired high over the bar.
The Congolese talisman again broke through the Ghana defence, only to slice his chance into the side netting.
Ghana showed a lot more intent in the second half, with Wakaso whipping a free-kick at Matampi, who saved with his feet.
Jordan Ayew - the younger of the two brothers - scored his first of the tournament, surging into the box and stroking the ball home confidently and decisively.
The lead lasted all of five minutes, with M'Poku unleashing a breathtaking long-range shot with the outside of the boot that arrowed into the top corner.
Ghana continued to press and were awarded a penalty when Christian Atsu went down in the box after contact from Lomalisa Mutambala, who was lucky not to receive a second yellow card.
Andre Ayew then coolly dispatched the spot-kick, sending Matampi the wrong way, to grab his second of the competition.
Leopards substitute Cedric Bakambu missed several chances to equalise in the final minutes, forcing a good save from Razak and finding space in the box at the death before shooting badly wide from close range.
Atsu could have finished it for Ghana in injury time, but the busy Matampi produced a great save at his feet.
The Black Stars' celebrations at the end showed Avram Grant's side's relief at edging the contest.
Match ends, Congo DR 1, Ghana 2.
Second Half ends, Congo DR 1, Ghana 2.
Foul by Dieumerci Mbokani (Congo DR).
Daniel Amartey (Ghana) wins a free kick in the defensive half.
Attempt missed. Cédric Bakambu (Congo DR) right footed shot from the centre of the box is too high. Assisted by Dieumerci Mbokani with a headed pass.
Foul by Cédric Bakambu (Congo DR).
Mubarak Wakaso (Ghana) wins a free kick in the defensive half.
Jonathan Bolingi (Congo DR) wins a free kick in the defensive half.
Foul by André Ayew (Ghana).
Corner,  Ghana. Conceded by Junior Kabananga.
Corner,  Ghana. Conceded by Ley Matampi.
Attempt saved. Christian Atsu (Ghana) right footed shot from the centre of the box is saved in the centre of the goal. Assisted by Thomas Partey.
Foul by Jonathan Bolingi (Congo DR).
André Ayew (Ghana) wins a free kick on the left wing.
Corner,  Congo DR. Conceded by Razak Brimah.
Attempt saved. Cédric Bakambu (Congo DR) right footed shot from outside the box is saved in the bottom right corner. Assisted by Dieumerci Mbokani with a headed pass.
Attempt missed. Cédric Bakambu (Congo DR) left footed shot from the centre of the box is too high.
Substitution, Congo DR. Jonathan Bolingi replaces Rémi Mulumba because of an injury.
Delay over. They are ready to continue.
Delay in match Rémi Mulumba (Congo DR) because of an injury.
Chancel Mbemba (Congo DR) wins a free kick in the defensive half.
Foul by Christian Atsu (Ghana).
Substitution, Congo DR. Cédric Bakambu replaces Paul-José M'Poku.
Substitution, Congo DR. Jeremy Bokila replaces Firmin Ndombe Mubele.
Foul by Issama Mpeko (Congo DR).
André Ayew (Ghana) wins a free kick in the defensive half.
Foul by Afriyie Acquah (Ghana).
Paul-José M'Poku (Congo DR) wins a free kick in the attacking half.
Goal!  Congo DR 1, Ghana 2. André Ayew (Ghana) converts the penalty with a left footed shot to the bottom right corner.
Penalty Ghana. Christian Atsu draws a foul in the penalty area.
Penalty conceded by Joyce Lomalisa Mutambala (Congo DR) after a foul in the penalty area.
André Ayew (Ghana) wins a free kick in the attacking half.
Foul by Issama Mpeko (Congo DR).
Daniel Amartey (Ghana) is shown the yellow card for a bad foul.
Foul by Daniel Amartey (Ghana).
Paul-José M'Poku (Congo DR) wins a free kick on the left wing.
Foul by Afriyie Acquah (Ghana).
Rémi Mulumba (Congo DR) wins a free kick in the defensive half.
Mubarak Wakaso (Ghana) wins a free kick in the defensive half.
Foul by Paul-José M'Poku (Congo DR).
Ghana's Ayew brothers grabbed a goal each to beat DR Congo 2-1 and put the Black Stars through to the Africa Cup of Nations semi-finals.