Rotherham, who missed a penalty at 0-0, always looked the more likely to win the match and they did so in extra time through Wood's second of the game.
Relive how the 90 minutes of action unfolded with Sports Mole's live text coverage below.
STARTING XI: Rodak; Emmanuel, Ajayi, Wood, Mattock; Taylor, Vaulks, Towell, Newell; Ball, Smith
SUBS: Price, Cummings, Purrington, Forde, Palmer, Williams, Lavery
STARTING XI: Henderson; Bolton, Sadler, Nsiala, Beckles; Godfrey; Whalley, Nolan, B.Morris, Rodman; C.Morris
SUBS: MacGillivray, Riley, Lowe, Brown, John-Lewis, Jones, Payne
Paul Warne: "I'm a pretty happy person so I'm not going there not intending to enjoy myself. When the lads warm up they will be hugely nervous but once you start the nerves dissipate and it is just a game on some different grass. We look at the best way we think we can hurt them and the best way we can stop them playing. The fact it is at Wembley and a play-off final does not affect the way we set up tactically. Shrewsbury know what to expect but they also know the disappointment of defeat. No doubt Hursty will have them pumped and ready to go."
Paul Hurst: "A lot of people would love to be in the position we're in now. We just hope there's one very special day left. We wanted to win the Checkatrade Trophy final, to get the chance to put our name in the history books. We wanted automatic promotion and we just fell short. When we perhaps lost our way slightly in the two semi-final games against Charlton, we stayed strong and put bodies on the line. But we have good people who work hard and commit to the shirt. And that's what we have to do again on Sunday."