Wood's clinical finishing at a rain-soaked Amex Stadium made the difference as Brighton was left frustrated by the heroics of Tom Heaton in the Burnley goal.
The Seagulls created more than enough chances to get a result but only found a way past Heaton after Ashley Barnes added to Wood's double when he scored a controversial penalty.
Shane Duffy grabbed a consolation for Brighton in the closing stages but Chris Hughton's side is still without a Premier League win in 2019, while Burnley is unbeaten since the turn of the year and sits 15th, three points clear of the bottom three.
Brighton survived a scare after 16 minutes when Shane Duffy diverted Dwight McNeil's shot onto his own crossbar and Ashley Barnes headed the rebound over the top, but Burnley goalkeeper Tom Heaton was soon called upon to deny a powerful Solly March effort with a one-handed stop.
The opener arrived after 26 minutes when Barnes floated a pass over the Brighton defence just as Lewis Dunk lost his footing, leaving Wood with a free run on goal and the New Zealand striker applied a clinical finish high into the net.
Heaton made a key intervention to push the ball away from Glenn Murray's head as the hosts pushed for an equaliser before half-time, with Jeff Hendrick then clearing off the line to preserve Burnley's lead at the break.
Brighton and Australia goalkeeper Mat Ryan will want to forget Burnley's second goal, which arrived just after the hour mark when McNeil surged forward with the ball and fed Wood, whose shot skidded off Ryan's outstretched palm on its way into the bottom-right corner.
Hendrick appeared to handle the ball in the Brighton box moments before Barnes was brought down by Ryan at the other end, and the Burnley forward confidently tucked home the ensuing spot-kick.
Duffy met Anthony Knockaert's long, looping free-kick over the top with a diving header that beat Heaton and gave Brighton a glimmer of hope that arrived far too late.