The Eels led 12-0 until the 67th minute before tries to Jason Ryles and Brad Takairangi tied it up at 12-12 and neither side could land a field goal in regular time.
Chris Hicks and Jarryd Hayne scored tries for Parramatta with Luke Burt booting two from two in front of 12,097 fans.
Anasta added two conversions to his field goal and Ryles' and Takairangi's tries for the Roosters' third straight win.
Parramatta led 6-0 at halftime and might have gone further ahead but for a Hayne long ball to Hicks going into touch, but the star's magic was soon back.
Wearing the No.6 jersey but back in the more familiar fullback position, Hayne finally recaptured some of his best form, beating three defenders and running 40m to the line.
The Roosters got it back to 12-6 when a controversial putdown from Ryles was not referred upstairs.
Burt missed a field goal shot in the 72nd minute and the Roosters locked it up through Takairangi two minutes later.
Anasta had three field goal attempts charged down, one soccered to the other end of the field by Hayne before the 12-12 stalemate after 80 minutes.
SYDNEY ROOSTERS 13 (J Ryles B Takarangi tries B Anasta 2 goals B Anasta field goal) bt PARRAMATTA 12 (J Hayne C Hicks tries L Burt 2 goals) in golden-point extra time at Parramatta Stadium. Referee: Jason Robinson, Grant Atkins. Crowd: 12,097.
- AAP