Chelsea have now won their opening two games to top WSL 1, while the Gunners have lost two of their first three WSL games and lie fourth.
England forward Kirby nipped in to loft over goalkeeper Sari van Veenendaal for her first, with Jemma Rose unable to prevent her shot crossing the line.
She then smashed in a second from three yards after a goalmouth scramble.
Chelsea lead the table on goal difference, ahead of Manchester City, who have also won their opening two games.
Liverpool will join the pair on six points from their first two outings if they beat Sunderland on Saturday.
Arsenal, for whom Van Veenendaal made a series of fine saves before Kirby's second settled the contest, will meet Chelsea again in the Women's FA Cup final at Wembley on Saturday, 14 May.
Arsenal 0-2 Chelsea as it happened
Chelsea forward Fran Kirby told BT Sport: "We came here to win the game. We want to win every game we can.
"We knew we had to get a result here and happily we did.
"The start of the season has been really good. Overall we're happy but obviously we just need to keep going."
Arsenal boss Pedro Martinez Losa told BT Sport: "It was a poor performance. We made it easier for the opponent.
"With performances like that we can't think about winning the league.
"Against a good team like Chelsea you can't make mistakes. We will try to come back, analyse and do better for the next game."
Fran Kirby scored twice as reigning champions Chelsea beat Arsenal to go top of the Women's Super League.