Spurs had lost seven of their last 10 matches at Wembley prior to Sunday's game, having been beaten 2-1 by Chelsea last time out thanks to Marcos Alonso's last-gasp winner.
But Mauricio Pochettino's men looked on track for a narrow victory thanks to Dele Alli's scrambled opener before Wood's late impact from the bench.
The striker, a club-record arrival from Leeds United earlier this week, scored the equaliser in the second minute of added time by thumping home a low finish from Robbie Brady's brilliant pass.
Alli had put Spurs ahead four minutes into the second half, smashing in a loose ball after the visitors failed to clear Christian Eriksen's corner.
But after Harry Kane, who has failed to score in 13 Premier League appearances in August during his career, wasted a string of chances, Wood came up with a priceless leveller for his side, who defeated Chelsea on the opening weekend at Stamford Bridge on their last away league outing.
1107 - Chris Wood has scored his first Premier League goal for 1107 days, last scoring for Leicester v Everton in August 2014. Equaliser.— OptaJoe (@OptaJoe) August 27, 2017
Set-pieces provided the only early chances as James Tarkowski and Ben Mee headed Brady corners over the crossbar in a bright start from the visitors. Kane should have done better with a Eriksen free-kick after 17 minutes, but the England forward could not get his header on target.
A series of crucial blocks maintained the first-half deadlock as Burnley frustrated Spurs, Tarkowski brilliantly deflecting another Kane shot wide to prevent a certain goal.
But the visitors' resistance ended as Spurs forced the breakthrough with the first shot on target of the match, four minutes after the restart. Mee failed to clear a right-wing corner from Eriksen and Alli reacted quickly to slam in the loose ball for his second Premier League goal of the season.
With Burnley on the ropes following the opener, Kane missed another fine chance, drilling wide of Tom Heaton's left-hand post after being found by Eriksen.
Burnley failed to muster a shot on target against West Brom and finally called Hugo Lloris into action after 70 minutes, the Spurs captain diving to his left to keep out a fierce Brady drive as the Clarets briefly threatened to equalise.
As the game opened up Heaton made two fine saves in quick succession, the goalkeeper denying his international team-mate Kane with a fine dive to his right and then reacting well to tip Eriksen's drive over the crossbar.
0 - Harry Kane's failed to score in every PL match he has played in August (13 games), despite attempting 42 shots in those games. Stunned. pic.twitter.com/7xDIZ87FzU— OptaJoe (@OptaJoe) August 27, 2017
Lloris raced off his line to deny substitute Wood and the striker then directed a header into the goalkeeper's arms in a late flurry.
And the New Zealand international sealed his stunning impact with a dramatic leveller, scoring his first Premier League goal since August 2014, timing his run on to Brady's brilliant pass to slam in a memorable maiden strike for the club in added time.
Key Opta stats:
- Spurs failed to beat Burnley in a home league game for the first time since October 1974.
- Harry Kane has failed to score in every Premier League match he has played in August (13 games), despite attempting 42 shots in those games.
- Spurs dropped their first points after going ahead in a Premier League game since losing 2-1 to Chelsea in November 2016 – they had won 21 consecutive matches when taking the lead before today.
- Chris Wood has scored 29 league goals since the start of last season – only Sheffield United striker Billy Sharp has scored more (34) in English league football in that time.
- The Clarets remain unbeaten away from home in the Premier League this season (W1 D1), the first time since 1973/74 they’ve avoided defeat in their first two top-flight away matches of a season.
- Spurs have now won only one of their last 11 matches played at Wembley (D3 L7).