The on-loan striker fired left-footed past Seagulls goalkeeper Tomasz Kuszczak in the first minute of stoppage-time following a powerful surge into the 18-yard box to give Stuart Gray's men all three points in a tight game at Hillsborough.
Albion went closest to breaking the deadlock in the opening stages when Spanish forward David Rodriguez, in search of his first Seagulls goal, headed Jake Forster-Caskey's left-wing corner against the Wednesday crossbar with 15 minutes played.
Owls winger Jeremy Helan then flashed a left-footed volley across goal at the other end after the Brighton back-line could only partially clear Best's delivery from the right.
The visitors were controlling possession and winger Will Buckley twice failed to pick out a team-mate after finding himself in promising positions deep in Wednesday territory.
The Owls, meanwhile, were reduced to a long-range effort from top-scorer Chris Maguire, which was charged down by full-back Stephen Ward just before the interval.
Buckley, who scored twice in his previous two games, was again at the centre of things 11 minutes after the restart, wasting a golden opportunity.
Forster-Caskey picked up the ball in the middle of the park before slipping it through the Wednesday defence for Buckley to beat the offside trap but, with only Chris Kirkland to beat, the former Watford man fired his shot straight at the Owls goalkeeper.
Jesse Lingard, who scored four against Wednesday while on-loan at Birmingham earlier in the season, was the next to threaten the home side's goal as Brighton almost caught them on the break in the 66th minute.
The on-loan striker scampered towards goal down the left-hand channel before cutting inside and striking a powerful effort from the edge of the box which was held by Kirkland.
Wednesday then went close to an opener with a rare chance at the other end.
Substitute striker Benik Afobe, a 71st-minute replacement for Helan, broke clear in the area but his stretched attempt was diverted away by Brighton keeper Kuszczak before Liam Palmer's powerful follow-up attempt whistled marginally over.
Kuszczak was again tested with 11 minutes remaining as Maguire sent in a stinging drive from distance.
Brighton had never won at Hillsborough in 13 attempts ahead of kick-off and Best confirmed that run would be extended when he collected the ball on the left, broke into the penalty area and drove home his fourth Wednesday goal of the season into the bottom right-hand corner.