Shay Logan’s goal was enough to see off the Dingwall men and County’s top goalscorer admitted his side were second best against the Dons in the fifth-round tie.
The winner came late on when the Aberdeen full-back’s shot took a slight deflection and went into the net off the inside of a post.
County goalkeeper Scott Fox had been the hero up to then with some fine saves, including an outstanding one to deny Dons skipper Ryan Jack early in the second half.
Boyce said: “To be honest, if anyone had chances in the game it was probably them. Scott Fox made a couple of good saves during the game. It was a bit of a scrap, a midfield battle. If I’m honest, we would have wanted to make more chances going forward.
“We had a good header from Jay [McEveley] but we were just unlucky. It was hard to take the way the goal went in, it was a deflection.”
Boyce, who has found the net 14 times this season, has not scored since the 6-2 cup win over Dundee United in the last round three weeks ago and, since then, County have only taken one point out of a possible nine, in a 1-1 draw with Rangers at Ibrox last week.
Boyce added: “The boys will regroup and focus on the league and try and get a few points on the board.
“We need to be more creative going forward and create a few chances. But with the players we have we can do that.”
Aberdeen’s match-winner Logan was delighted with his decision to take a pop with his ‘wrong’ foot late on.
He said: “I’ve scored a few goals with my left foot but it took a nice deflection on its way into the net.”
Logan acknowledged it wasn’t easy against a well-drilled County side, adding: “They are a tough team, they’ve got some big players.
“But we knew we had the ability to match that – it is all about concentration. We got a team performance today albeit so late, we will take a 1-0 in the 90th minute.”