The Royals had won a replay at Liverpool in the third round - after Sigurdsson netted a penalty to level the tie in stoppage-time - to earn another crack at a top-flight side.
McDermott's men were rewarded for a battling second-half display when, with just three minutes left, Andy Griffin's deep ball found its way through to the six-yard box where Sigurdsson guided his shot into the bottom corner for a ninth goal of the season.
The 20-year-old had shaken off a foot injury to start, and McDermott was full of praise for the young Icelander.
"Gylfi had a couple of other chances before he scored, but his finish for the goal was from the top drawer," said McDermott.
"That is what he has in his locker.
"We were waiting for him to grow while he was in the Academy. He came over here when he was about 13.
"He is getting bigger and has a real aura around him, especially since he scored that goal against Liverpool, but he is a real down-to-earth lad."
Reading - relegated from the Premier League in 2008 - are struggling at the wrong end of the Championship table, with McDermott in temporary charge following the dismissal of Brendan Rodgers.
Chairman John Madejski is set to make a decision shortly. Currently away on business, Madejski may have been affected by technical problems affecting a live feed of the game on the FA's official website.
"I have said all along that I want the job and that I'm the right guy for it at this moment in time," said McDermott, who has been at the club for the best part of a decade since joining as chief scout in September 2000.
"We will see what happens when the chairman comes back next week.
"He will have watched the game on the website, but if it crashed I will have to give him a ring!
"I would like a home draw. Someone said we it would be nice to draw Manchester United - and I said 'we would do well to beat them'."
Despite their cup heroics, McDermott knows consistent success in the league is the priority.
He said: "The situation we are in we have to win games, and even though this was not a league game we wanted to get through to the next round.
"There is not a lack of confidence in this side. You can see we have a belief.
"We are just trying to build slowly and trying to make something of the season.
"We need to keep performing and if they show the spirit they have done today, then the results will come."
Burnley manager Brian Laws lamented his team's failure to make the most of their dominant spell midway through the second half.
"We desperately wanted to get through to the next round, and we did not play as well as we were capable of," said Laws, who was in charge for only his second game since being appointed as Owen Coyle's successor.
"[Steve] Thompson up front had given it a go, even though he was sick during the night, then in the second half when we changed it with [Martin] Paterson we showed a bit more energy.
"We upped the tempo and looked much more dangerous.
"We had a good 20-minute spell in the second half and looked more like ourselves, passing it better.
"We created a few good chances, with some quality saves from their keeper which kept Reading in the game.
"We knew that if we did not take our chances, we were going to come under some pressure and the disappointing thing is that we were only three minutes from getting our first clean sheet in quite a while."PA Sport