They were made to sweat during the second half as Marcelo Bielsa’s side put them under severe pressure.
But the Toffees held on to bounce back from Saturday’s disappointment against Newcastle in fine style.
Carlo Ancelotti made four changes to his side, with Robin Olsen coming in for the injured Jordan Pickford.
Seamus Coleman, Michael Keane and James Rodriguez dropped to the bench, with Mason Holgate, Ben Godfrey and Andre Gomes coming in.
It looked like Carlo Ancelotti was seeking to match Leeds’ energy after they outran and outperformed the Toffees at Goodison earlier this season.
He was rewarded with an opening goal after just nine minutes.
Andre Gomes searching pass found the onrushing Lucas Digne and his low cross found Sigurdsson, who had ghosted into the area beyond three Leeds defenders and prodded home.
The nearly doubled their lead moments later when the ball broke for Calvert-Lewin on the edge of the box, but his left-footed shot was easily saved by Illan Meslier.
That appeared to spark Leeds into life, who grew into the game and created a flurry of chances.
First, a Mateusz Klich curler from range was comfortably punched away by Olsen.
Ezgjan Alioski then nearly scored a goal of the season contender when he met a corner with a first-time volley on the edge of the box that smacked off the post (though replays suggested Olsen may have had it covered).
Pascal Struijk then met Kalvin Phillips free-kick with a header that was tipped over by the Toffees goalkeeper.
Everton responded, though they weren’t helped by a slippy surface that was only relaid a few days ago, with players from both sides slipping all over the place.
Sigurdsson had an effort from distance saved by Meslier before Calvert-Lewin earned a rollocking from Carlo Ancelotti for opting to shoot instead of passing to the onrushing Richarlison.
He didn’t have to regret that miss for long.
Just before half-time, Sigurdsson’s corner was flicked on by Ben Godfrey and Calvert-Lewin was there at the far post to head home his first Premier League goal since November.
What don’t you do when you’re 2-0 at half-time?
Conceded right at the start of the second half.
What did Everton do?
Conceded right at the start of the second half
Some comedy defending in the area as Yerry Mina and Mason Holgate failed to clear, allowing Raphinha to side foot the ball into the corner.
Game on. And having been to Elland Road before, I was glad the game was behind closed doors and their boisterous supporters were unable to roar their team on.
Everton thought they should have had a penalty when Luke Ayling handballed in the area but the referee and, puzzlingly, VAR, said no spot-kick.
Olsen then earned his wages with three stunning saves in the space of a minute, first denying Klich and Raphinha before parrying away Jack Harrison’s effort.
Richarlison, who looked desperate for a goal, cut in from the left and curled in an effort that was never troubling Meslier, but it at least stemmed the White tide momentarily.
And it was momentarily. A neat bit of interplay by Leeds resulted in Stuart Dallas crossing for Patrick Bamford, with his looping header clipping the top of the crossbar.
Ancelotti switched things up with 10 minutes left. A rather grumpy Richarlison slumped off and was replaced by Michael Keane, with Godfrey switching to left-back and Digne pushed forward.
The Toffees had the chance to finish the game when the superb Abdoulaye Doucoure led a three-man counter and played in Calvert-Lewin, but his low shot was well saved by Meslier.
Seconds later deadline day signing Josh King got his first taste of life in a blue shirt, coming on for Iwobi.
The Blues managed to see out the game to continue their fine form on the road and lift themselves back into European contention.