In a game that saw very few chances in the 45, Dele Alli stepped up to open the scoring four minutes into the second-half to give the hosts' the crucial breakthrough.
Spurs missed plenty of chances to add to their lead and they were punished in the second minute of injury time when Wood scored on his Premier League debut to earn a point for Sean Dyche's hardworking side.
Once again though Mauricio Pochettino will be left thinking what do his side have to do in order to win at the National Stadium, where they always seem to struggle playing.
Spurs' were left frustrated during the first half by dogged Clarets
In terms of the game itself, the first-half had very little goalmouth action at all with the visitors defending very well to not give the the hosts' any space to do what they do best.
It worked well as Tom Heaton only had one save to make when Heung-Min Son was given the ball, his shot stinging the palms of Heaton.
At the other end, Robbie Brady forced Hugo Lloris into a very smart save, but apart from that the half was very scrappy as both teams struggled to create much space at all.
Therefore, at half-time, Pochettino would have been looking for much more from his side as they needed to somehow play much better to try and get an important win.
Alli opened the scoring for Spurs' early in the second half
It seemed to have the desired effect as it didn't take long for them to open the scoring after the break. Alli managed to put the ball into the back of the net at the second attempt of asking after a good cross from Christian Eriksen was flicked onto the midfielder's path by Eric Dier.
It was very nearly two a couple of minutes later when Eriksen gave the ball to Harry Kane in the box but he saw his curling shot go just wide of the post much to the frustration of the striker, who still couldn't score in the Premier League in August.
At one-nil though Spurs were never comfortable in the game and they were given a warning sign when Brady forced Lloris into a very good save at his near post just after the hour mark.
Kane had another chance to make it two for Spurs soon after the Brady chance when he was played in one on one against Heaton but again the 'keeper came out on top.
Wood punishes wasteful Spurs in injury time to earn his side a point
The Spurs fans were very restless in the stadium right at the end and they had every right to be as in injury time, Wood, on as a second-half substitute, received the ball in the box from a fantastic Brady pass and he calmly slotted the ball into the back of the net to earn the Clarets a very important point and once again stop Spurs from winning at Wembley.