The Bears bowled first at Edgbaston Stadium and made early inroads, reducing Essex to 27 for four, with Amu Surenkumar striking three times inside the first six overs.
A 56-run partnership between Jo Gardner (33) and Liberty Heap (23) steadied the ship for the visitors, but when Bears captain Georgia Davis dismissed both players in the space of three balls, Warwickshire were able to restrict Essex to 116 for eight from their 20 overs.
In reply, the Bears lost Katie George and Nat Wraith early, and were at one point 61 for four, but Redmayne played a starring role with the bat, ending unbeaten on 43 from 41 balls to safely see the hosts home, five wickets down.
“I probably gave them a few chances to get me out but it’s certainly nice to be there at the end,” she said.
“Sometimes those small targets can be a little bit nervy. We certainly lost more wickets than we would’ve liked and it probably wasn’t quite as convincing with the chase, or as clinical as we would’ve liked, but we got over the line.
“To get three wins in a row is really special, and it’s nice to build some confidence going into the back end.”
The Bears, who had lost their opening six fixtures in the competition, have bounced back in emphatic fashion, getting up and running with a 42-run win against high-flying Surrey and then following it with a nailbiting one-run win over Lancashire prior to today’s victory.
Redmayne says momentum can be crucial in T20 cricket, especially with Blast fixtures coming thick and fast.
“The schedule over here it’s very different to back home in Australia,” she added. “It’s fast-paced and there’s not too much time to regroup between games, so you really have to ride that momentum in a way.
“That win against Surrey was really special. They’re sitting near the top of the table, so to come from the bottom and to take on one of the top teams and beat them at home was really nice. We took a lot of confidence from that and sometimes you just need one win to get you going.
“We’ve got quite a few home games as well now. Edgbaston is a great place to play. It’s certainly exciting and hopefully against Yorkshire on Sunday we can go four from four.”
Bears Women return to action against Yorkshire at Edgbaston on Sunday 5 July as part of another home doubleheader. Play starts at 11:30am, with Bears Men playing against Gloucestershire at 3:30pm.