With the npower League One strugglers 1-0 down Jose Baxter hit the post against his former club with a 25-yard lob before their penalty claims fell on deaf ears when Phil Jagielka's wayward clearance struck Darron Gibson's arm.
Leighton Baines and Leon Osman then gave Everton a healthy lead before Matt Smith's consolation but Furman admitted things could have been different.
"On nights like this you need all the luck - you need that shot to hit the inside of the post and go in rather than hit the inside of the post and trickle across the line," he told Press Association Sport.
"There might have been a penalty straight after that. For us, we need those kind of decisions to go our way.
"It wasn't to be but we're proud of what we did. It was a great cup run for the players, we've gone and played some great teams, but also for the fans to have the opportunity to come out here and enjoy the night and the financial aspect of the FA Cup run is brilliant for the club."
Oldham were applauded off the pitch by all four sides of Goodison Park after a battling second-half display and Furman was encouraged by the fighting spirit they showed in the second 45.
The South African said: "We were a little bit disappointed with out first-half performance, we never really got to grips with it and one of the things that was said at half-time was 'do we just want to go out with a whimper, without a fight?'
"But we made it difficult for them in the second half and I think we gave our great support something to sing about.
"We caused them problems in the second half and we can take credit from that."
- Sports & Recreation
- Phil Jagielka
- Darron Gibson