Manager Craig Brown felt his Aberdeen side were worthy winners after they ended Motherwell's interest in the William Hill Scottish Cup for a second successive season.
Rory Fallon, scorer of both goals in the Scottish Cup quarter-final win at Fir Park last season, netted the 51st-minute opener and Joe Shaughnessy tapped in 10 minutes later, as the Dons won the fourth-round replay 2-1 to set up a fifth-round clash with Hibernian at Easter Road.
Motherwell, the 2011 runners-up, had lost just once in 13 previous games against Aberdeen, but Brown said: "We've now managed to turn it round against Motherwell. That's four games and it's two wins and two draws for us."
He added: "Between the first and the second goal we had three really good chances and (Motherwell goalkeeper Darren) Randolph had two saves in the game which were world class."
Hibs beat the Dons in last season's semi-final, but Brown's men will be hoping to repeat last month's 1-0 Clydesdale Bank Premier League at Easter Road to progress.
"It's another testing fixture. We've got to rise to the occasion again," Brown said. "We've done very, very well away from home, so an away game is not a disadvantage to us."
The absence of captain Russell Anderson (back) forced a defensive reshuffle for the Dons, but youngsters Clark Robertson and Shaughnessy, who signed two-year contract extension last week, stood up to the challenge, while Andrew Considine and Mark Reynolds played well.
Brown added: "We handled the situation exceptionally well, particularly the young boys out there. I thought they were superb. To come here without Russell and to get a victory against Motherwell is very good. I rate Motherwell highly. I rate (Motherwell manager) Stuart McCall and I think they have a superb team."
Michael Hidgon's stoppage-time penalty was merely a consolation for the hosts, who might have had a spot-kick in the first half. Jamie Murphy moved into the area and a clumsy Shaughnessy stumbled into his back, but no penalty was awarded.
McCall, though, had no complaints. He said: "I'm disappointed to be out of the cup. It wasn't to be. Second half they were better, there's no doubt about that. We couldn't find a way past them. I thought Reynolds was outstanding at centre back for them."