The game looked beyond them when they went 3-1 down in the first 18 minutes, but Ward hit her second goal just before the interval.
Then, after Jodie McGuckin equalised, the Blues striker hit the winner with a superbly struck 30 yard free-kick 16 minutes from time.
"It's a weekend I'll never forget, that's for sure," said manager Marcus Bignot.
"The girls perhaps lacked a bit of self-belief early on today, but I went up there believing we would win the game and the girls believed it too once we had got to half-time.
"This is the greatest day in the history of the Birmingham City women's team - but hopefully there are more days like this to come."
The girls will now meet the winners of next Wednesday's semi between Fulham and Arsenal.
The final is to be played at Wycombe Wanderers' Adams Park on 10 March.
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