Manager Jim McIntyre admits Ross County will have to play a lot better if they hope to repeat the feat of 2010 and get to the Scottish Cup final.
Substitute Rafa De Vita scored the only goal of the replay against Dunfermline 20 minutes from time.
However, it was a far from convincing performance from the Premiership side against the League One leaders.
"I'm delighted to get through but we haven't played well in the two games against Dunfermline," said McIntyre.
County's visitors certainly did not look out of place against the Staggies and McIntyre admits progressing to a home tie against Linlithgow Rose or Forfar Athletic was down to grit and determination.
In fact, the former Pars manager was more than happy to concede his old club were unlucky not to be the ones who made it through to the last 16.
"What we showed tonight was a spirit and somehow we managed to get ourselves through to the next round of the competition," McIntyre told BBC Scotland.
"Great credit must go to Dunfermline as they were definitely the better side in the first game and were unlucky not to progress on Saturday.
"We have taken advantage of that luck tonight. They have got good players but from our point of view I haven't seen us so wasteful with possession for a number of weeks.
"We were just making poor decisions at the wrong times as when it was on to take a touch we never and when it was on to make a pass we were getting caught on the ball.
"Of course we didn't play well tonight but we got through to the next round and that's what matters in the cup.
"We know we will be favourites in the next round whoever we play but we will do our homework on them as we will show them every respect."
Defeat for Dunfermline was their first defeat in 14 games since losing to St Mirren in the Challenge Cup in October.
But they are still nine points clear at the top of League One and while promotion was always top priority manager Allan Johnston admits his players are disappointed to be knocked out of the cup.
"Our boys ran Ross County very close," said Johnston. "I don't think we could have done much more as we looked like the Premiership team.
"The way we passed the ball, the chances we created and the pressure we created was good and it sums it up that [Ross County goalkeeper] Scott Fox is lying on the ball trying to waste time at the end.
"The players deserve a lot of credit for a terrific team effort as they kept going. They don't know how to get beat and expect to win every game.
"Obviously it is a big blow getting beat so we have to lift ourselves for the league game at the weekend [against Albion Rovers].
"They are gutted after the way they have played but gone out and that shows you about the confidence that we expect to come to Ross County and win."