Former Newcastle star Pollicina set Saturday's game alight when she curled a wonderful long-range strike into the top corner for her first City goal in the 38th minute.
Two minutes later, Pollicina turned playmaker, looping in a tantalising corner.
Wilkinson, who hadn't troubled the scorers in her previous two games, rose highest to direct her header into the back of the net.
"That's why we brought (Pollicina) here," City coach Rado Vidosic said.
"When she was at the Jets, she had the freedom to go anywhere she likes. Here it's more about structure.
"I'm so, so happy for her that finally she performed really, really well and then as reward was a beautiful goal."
"For (Wilkinson) to score a goal (that brought her) confidence up.
"Now the confidence will be through the roof. So it's a perfect timing for the Melbourne Victory derby (next week)."
Adelaide were without skipper Isabel Hodgson through injury, with Emily Condon taking the captain's armband.
Despite Condon being forced from the field herself with a corked quad just before the hour, Adelaide sparked the game back to life in the 65th minute.
From a corner, Shadeene Evans put a header on target and City goalkeeper Melissa Barbieri parried the ball into Chelsie Dawber's legs, before the forward instinctively toe poked home.
Adelaide appealed for a penalty with four minutes remaining in regulation time when Fiona Worts went down just inside the area under contact from City skipper Emma Checker, but it wasn't forthcoming.
"For me it was a clear penalty. I haven't seen a replay of it," Adelaide coach Adrian Stenta said.
"But the only thing I can say, and it's not me that's making the decisions, is that all we asked for as coaches is consistency and I think that exact same foul all over the park got called two or three times today."
City pounced on Adelaide's defence in added time, with Leticia McKenna putting in a cross for Wilkinson to coolly finish from close range.
"I'm sitting here feeling really frustrated because I think we were good enough to get something out of that game coming into it, but I don't think we showed our best football, particularly early on," Stenta said.
"In saying that I'm pleased that again, this team has shown a little bit of resilience. The challenge for us as a group is to do that from the start."
City play Victory on Boxing Day while Adelaide were scheduled to play Perth Glory on December 23 but now look unlikely to be in action again until the new year given Western Australia's border restrictions.
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