Arsenal had the better of the remainder of the second half but forced Brown into only save, an 88th-minute goal line stop from midfielder Jennifer Beattie's header.
And five minutes into extra-time Everton should have regained the lead, right winger Jody Handley finding Dowie free but from eight yards she shot feebly at Byrne.
As extra-time progressed play switched rapidly from end to end and in the 109th minute only a great block by right-back Becky Easton prevented central defender Gill Flaherty from putting Arsenal in front.
A penalty shoot-out was beckoning when Dowie raced on to substitute Brooke Chaplen's pass, dinked the ball over the advancing Byrne to score the winner.
It was her 28th and most important goal of the season, coming on the day when her uncle saw his Hull side relegated from the Premier League.
The Toffees now face the Gunners again on Sunday, this time in the league when a victory for the Londoners could wrap up the title.
And Arsenal's disappointed manager Laura Harvey said: "We had enough chances to have won, but now we have to focus on winning the league - and to do it by beating Everton would be great."
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