The gripping clash at the City Ground on Monday night, ended 1-1 after extra time, setting the stage for a nail-biting finish that sent Forest through to face Brighton next.
The hosts, currently soaring in third place in the Premier League, have been the season's surprise package, but it was relegation-battling Ipswich who struck first.
Eight minutes into the second half, George Hirst rose to nod home a back-post header, silencing the home crowd. Forest responded midway through the half when Ryan Yates met Anthony Elanga's pinpoint cross to head in his fourth goal of the campaign, restoring parity.
Chances came thick and fast thereafter. Yates thought he had doubled his tally minutes later, only for the effort to be ruled out, while Callum Hudson-Odoi rattled the crossbar with a venomous strike from distance.
Extra time saw both teams trade blows in an end-to-end spectacle, but neither could find a winner, forcing the tie into penalties.
With all nine prior kicks converted amid a cauldron of noise, Sels stepped up, thwarting Taylor to spark wild celebrations. "It was the only penalty I picked the right corner for!" Sels said after the match. "In a shoot-out, one goalkeeper has to be the hero. I'm delighted it was me. Now we look ahead to the weekend."
For Nuno Espirito Santo, the victory marked another step toward a dream Wembley trip. The Forest boss admitted the shoot-out tested his nerves. "I'm really happy," he said. "The FA Cup means so much to us, and penalties bring huge emotions. The players executed well, and Matz was amazing again. It's a big moment for the club, we're giving this competition a real go."
Nuno revealed the team stuck to their penalty blueprint from the previous round, where Sels also shone in a shoot-out win over Exeter. "We've practiced after every FA Cup draw," he added. "It's about preparation and dedication, getting the order right and giving the goalkeeper as much info as possible."
Ipswich, now out of the cup, turn their focus to Premier League survival with 11 games left. Manager Kieran McKenna praised his side's resilience despite the heartache. "It's tough to lose that way," he said. "But I'm proud of the players, they left everything out there. We competed with one of the league's best and can take confidence from that into our next fight."
Forest's social media buzzed post-match, with the club posting on X: "Matz Sels, take a bow, into the quarters we go!" Sels himself chimed in, tweeting: "What a night at the City Ground. Onward and upward!" Meanwhile, Ipswich's X update reflected defiance: "Heads high, now we battle in the league."
The Reds' FA Cup journey rolls on, with Nuno's men proving their mettle - and their spot-kick prowess - once again. For Ipswich, it's back to the relegation scrap, but with renewed belief after pushing a top side to the limit.