American forward Woods notched a goal and an assist in just her third ALW game as Brooke Nunn scored twice against the Reds at Porirua Park.
Later on Sunday, Sydney FC goalkeeper Heather Hinz played a starring role as a wasteful Melbourne Victory was made to settle for a goalless draw in the Big Blue.
Woods has hit the ground running since arriving at the Phoenix in January as an injury replacement for Dutch midfielder Tessel Middag (ACL).
Relatively unknown, Wellington coach Bev Priestman had taken a punt on Woods after being impressed by the American's college career.
The American forward opened Wellington's account on Sunday in the 15th minute after being on the receiving end of a cutback from 'Nix midfielder and former Reds player Lucia Leon.
Nunn added to the hosts' lead three minutes later, firing from the tightest of angles past goalkeeper Illona Melegh after Woods threaded the ball through Adelaide's defence.
Woods was substituted off in the 78th minute after struggling with what looked like a cramp in her left leg, before Nunn completed her brace in the 86th minute.
The Reds got a consolation goal a minute later through Isabel Hodgson, whose shot looped over the head of goalkeeper Victoria Esson.
Celebrating midfielder Manaia Elliott's 50-game milestone in style, Wellington (5-3-3) moved from eighth to second to continue their rise under former Canada women's boss Priestman.
The Phoenix had finished ninth last season after just seven wins from 23 matches under Paul Temple.
After starting round 14 in fourth, Adelaide (17 points) drop out of the top six after Melbourne Victory's draw with Sydney FC, given Victory's superior goal difference.
Spearheaded by American striker Kennedy White, Victory settled for a point after firing off 23 shots against the Sky Blues' 12 at Casey Fields.
Both sides were forced to make midfield changes through injury, with Sydney without Bianca Galic (ankle) and Rhianna Pollicina (wrist) missing for Victory.
White should've broken the deadlock in the 57th minute after plucking the ball off Sky Blues centre-back Willa Pearson to set up a one-on-one with compatriot Hinz.
But shotstopper Hinz read the play better, intercepting White's shot with her outstretched foot.
White tried her luck again from close range in the 87th minute to no avail, with a diving Hinz getting her glove on the ball.
Victory have gone five games without a win but at least snap a two-game losing streak, while Sydney FC languish in 10th (11 points).
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