An unmarked Sigurdsson steered Lucas Digne’s outstanding low cross from the left flank for the first time into the net, the visitors took a ninth-minute lead.
Ezgjan Alioski volleyed against the post, although Leeds came close to equalizing and Everton goalkeeper Robin Olsen acrobatically tipped over Pascal Struijk’s header.
After Ben Godfrey had flicked on Sigurdsson’s left-wing corner, with a close-range diving header England striker Calvert-Lewin grabbed his 12th Premier League season goal.
Raphinha pulled one back from Patrick Bamford’s pass with a calm finish, but the hosts couldn’t catch an equalizer and Leeds were the better side in the second half.
The victory moves the Toffees, who started the day eighth, above Chelsea, Tottenham, and West Ham, although the Hammers will regain fifth if they beat Aston Villa in a later game on Wednesday.
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.