Everton defender Vitaliy Mykolenko put through his own net late in the opening 45 minutes, unable to get out of the way as Alfie Doughty's delivery from a corner arrived at the near post, to round off a turgid first half of football in which both sides struggled to create openings.
Despite getting back in the tie, Everton were unable to shift the momentum in their favour as Luton carved out the better opportunities, with Elijah Adebayo forcing a fine save out of goalkeeper Joao Virginia and Carlton Morris seeing a shot cleared off the line by Nathan Patterson.
With the minutes ticking on and Premier League relegation rivals Everton and Luton both keen to avoid a replay, spaces started to appear with greater regularity but, as was the story of the first half, both sides were wasteful.
That was until the 96th minute, when Woodrow volleyed in from close range after a goalmouth scramble when Everton failed to clear their lines from a corner.
It is just the second time Luton have reached the fifth round of the FA Cup since 2012-13, when they were in the Conference Premier - now known as the National League.
More to follow.