Goals either side of the break secured a comfortable win for the visitors to end Bristol's 100 per cent start to the season end.
“That’s more like the sort of Birmingham performance we are used to and I was delighted with our display,” said Blues Ladies boss David Parker.
Williams headed home from Jade Moore’s cross on the half-hour and Linnett crashed home from close range following a corner 20 minutes into the second period.
“We know Rachel is fantastic in the box and powerful and strong in the air,” said Parker.
“All the girls are delighted for Kirsty because at one stage it looked like her career might be over. She has come back from two serious anterior cruciate injuries and it has been a long, dark road for her at times.
“Hopefully this is a result that will kick-start our season and we can push on from here.”
Bristol had gone into the game looking to make it five wins in a row, but it was the Blues who made the better start.
Chelsea Weston’s cross from the right after 10 minutes just evaded the leaping Jo Potter at the far post, with the Birmingham striker looking certain to score had she managed to make contact.
Nine minutes later, it needed an excellent diving block by Bristol defender Grace McCatty to prevent Izzy Christiansen’s fierce shot getting through.
It was midway through the opening period before the Vixens created anything of note, but Nikki Watts’ cross from right was headed over by Natalia Sanchon.
Shortly afterwards, the Spaniard let fly with a spectacular volley from the edge of the box. She had her back to goal when she met Loren Dykes’ cross, but still forced Birmingham keeper Mary Earps to grasp the ball just under the bar.
The visitors broke the deadlock on the half-hour when Jade Moore delivered a cross which provided Williams, who had lost her marker, with a relatively simple chance to head home from six yards.
Bristol tried in vain to respond, but Corinne Yorston’s low, driven cross from the left was smothered by Earps and then Watts shot into the side netting from Natasha Harding’s deep cross.
Birmingham killed the game off after 65 minutes when Potter’s corner from the right was played back across goal by Christiansen and Linnett smashed her shot high into the net from close range.
The Blues could have added a third in the closing stages when a ball over the top put Williams clear, but keeper Siobhan Chamberlain did just enough to divert the ball away from the striker as she closed in on goal.
Bristol have never beaten Birmingham and Vixens’ boss Mark Sampson admitted his side were well below their best.
“The better side won and Birmingham were probably better than us all over the pitch,” said Sampson.
“When that is the case you are not going to get a result and it has taught us a lesson that we can’t get ahead of ourselves. It was a very disappointing night for us and we need to respond on Sunday in our home game against Lincoln.”
Bristol Academy: Chamberlain; Dykes, Rose (Matthews 69), McCatty, Yorston; Harding, Windell (James 69), Staniforth; Watts, Sanchon, Del Rio (Curson 69). Subs (not used): Hignett, Cleverly.
Blues: Earps; Weston, Bassett, Harrop, Unitt; Lawley, Moore, Potter, Christiansen; Linnett (Haines 70), Williams. Subs: Hourihan, Simpkins, Torkildsen, Wilkinson, George, Westwood.