The Toffees lost to their Championship hosts 8-7 on penalties after the game ended 2-2 following 90 minutes.
But it's QPR who are in the hat for the fourth round as they won the shootout, with Ben Godfrey and Tom Davies missing spot-kicks for Everton.
Manager Rafa Benitez told BBC Merseyside: “It's never good to go out, it doesn't matter how. But this way, with the team giving so much. We fought back twice to stay in the game and in the second half, we created a lot of chances.
“Normally, you expect you can win. We didn't do it and after the penalties, sometimes it is luck.
“Overall, the team was giving everything, the attitude was very good and it's a pity. In both areas, we maybe could have done better.
“We were pushing and it's a pity because the team was doing everything to win the game. We didn't take our chances and in the second half, we were in control most of the time.
“The way we were playing and the attitude of the players, we were trying to go through but that's football. You make mistakes and pay for that.”
To make matters worse, Digne and Andre Gomes suffered injuries at the Kiyan Prince Foundation Stadium. Both came off before the game was over.
Everton already have four key players in Richarlison, Seamus Coleman, Dominic Calvert-Lewin and Jordan Pickford sidelined ahead of Norwich's visit to Goodison Park in the Premier League on Saturday.
Benitez is hopeful Digne and Gomes won't be joining the quartet on the treatment table.
He said: "We have a couple of issues. We will see what happens with Andre Gomes and Lucas Digne.
“I suspect it won't be too serious but we'll have to see how they are tomorrow."
‘We rode our luck’
QPR boss Mark Warburton was delighted to take the scalp of their top-flight visitors.
However, he admitted his side rode their luck at times.
Warburton said: "I'm delighted, especially with the first-half performance. I thought we were on the front foot, tenacious, played forward quickly and deserved the lead at half-time.
“In the second half, we rode our luck, if I'm honest. They came onto us, which I knew they'd do and they have quality players, but we rode our luck.
"But it comes to a penalty shootout and it went our way.”