When an unmarked Sigurdsson steered Lucas Digne’s excellent low cross from the left flank for the first time into the net, the visitors took a ninth-minute lead.
Leeds came close to equalizing, but Ezgjan Alioski volleyed against the post and acrobatically tipped over the header of Pascal Struijk by Everton goalkeeper Robin Olsen.
England striker Calvert-Lewin then grabbed his 12th Premier League season goal with a close-range diving header after Ben Godfrey had flicked on Sigurdsson’s left-wing corner.
But Leeds were the better side in the second half, and Raphinha pulled one back from the pass of Patrick Bamford with a cool finish, but the hosts were unable to catch an equalizer.
The win puts the Toffees, who began the day eighth, above Chelsea, Tottenham, and West Ham, but if they defeat Aston Villa in a later game on Wednesday, the Hammers will regain fifth.
Leeds United – Everton 1-2
Goals: G Sigurdsson (9), Raphinha (48), Calvert-Lewin (41)
Leeds: Meslier, Ayling, Struijk, Cooper, Alioski (Hernandez 78), Phillips, Raphinha, Klich (Roberts 69), Dallas, Harrison (Costa 87), Bamford.
Everton: Olsen, Holgate, Mina, Godfrey, Digne, Doucoure, Andre Gomes, Iwobi (King 87), Sigurdsson (Davies 89), Richarlison (Keane 80), Calvert-Lewin.