Champions Chelsea slumped to a 2-1 defeat away to newly promoted Liverpool last week, but Kirby's first-half strike and a second-half penalty from Mjelde was enough to get them back in the win column against City in front of more than 4,000 fans at Kingsmeadow.
That crowd figure was dwarfed at the Emirates Stadium on Saturday as Arsenal and Spurs clashed, with Vivianne Miedema scoring twice as the Gunners registered their second straight 4-0 league win to top the standings.
Brighton & Hove Albion also got their first win at the second attempt, beating Reading 2-1 to send them to the bottom of the table, while Manchester United climbed into second place with a comfortable 2-0 win away to West Ham United.
They are level on six points with Arsenal and Aston Villa, who continued their good start to the season by beating Leicester City 2-0.
Liverpool host Everton in the season's first Merseyside derby later on Sunday.
