Wood's opening goal was cancelled out by Alex Rodman an hour into the match, but his second came too late for Shrewsbury to respond for a second time.
Rotherham return to the Championship after a season-long absence, while for Shrewsbury - who relied heavily on keeper Dean Henderson - their 29-year wait for second-tier football goes on.
Paul Warne took the decision to make three changes to United's starting lineup, and it was his side who had the better of things in the first half, despite finishing eight points below Salop at the end of the regular season.
After a couple of early crosses into the box, Omar Beckles was grateful that the referee waved away penalty appeals after one clearance popped up and hit him on the arm.
The centre-back was not so fortunate two minutes later, though, as he was penalised for dragging down Wood inside the box.
Highly-rated Manchester United goalkeeper Henderson, on loan with the Shrews for the season, was equal to David Ball's spot kick, however, giving his side a temporary reprieve.
Town were struggling to find any real rhythm, perhaps still haunted by their defeat in the EFL Trophy to Lincoln City here last month, with Alex Rodman's blocked shot and Carlton Morris's flicked header their only real sights of goal in the opening 45 minutes.
Despite also not being at their best, Rotherham took a deserved lead into the interval after Wood again won his battle with Beckles by getting his head to a corner first to glance home.
Shrewsbury had a little more about them in the early parts of the second half, and they were level from the game's next big chance just short of the hour mark.
Following a low-quality period of 20 minutes, the tie finally burst into life in the final quarter of normal time as both teams created some chances.
Lavery could have made himself a hero when played through on goal 12 minutes from time, only to curl the ball a yard wide with just Henderson to beat.
Having been kept quiet for large parts, Michael Smith was the next player to squander a glorious opportunity as he got the ball stuck under his feet when attempting to round Henderson late on.
Whalley and Ryan Williams exchanged shots, both of which were easily dealt with by Rodak and Henderson respectively, but the latter would be called into action one last time to keep out Williams with a fine save from the final act of normal time.
The match-winning moment would arrive towards the end of an otherwise quiet first period of extra time, with Wood this time getting in front of John-Lewis and converting with a first-time finish.
Victory for Rotherham ensures that all three of last season's relegated clubs have returned to the Championship at the first attempt - the first time that has happened at any level in the English Football League since 2006-07.
ROTHERHAM UNITED (4-4-2): Rodak; Emmanuel, Ajayi, Wood, Mattock; Taylor (Williams 76'), Vaulks, Towell, Newell (Forde 111'); Ball (Lavery 71'), Smith
SHREWSBURY TOWN (4-1-4-1): Henderson; Bolton (Riley 73'), Sadler, Nsiala, Beckles; Godfrey; Whalley, Nolan, B.Morris (Payne 56'), Rodman; C.Morris (John-Lewis 66')