But a run of 10 wins from their last 12 games has put them well on course to end that two-decade wait for continental competition.
Forest became just the second side in Premier League history to respond to losing a game by five or more goals by winning by that margin.
“That is exactly what we wanted after last week, and exactly what the fans wanted,” said Gibbs-White.
“Last week was no way near our standards and we have been working on things to put it right all week in training, and the lads were incredible today.
“We completely nullified everything they did and caught them a couple of times on the counter. We were clinical in the final third.”
Brighton boss Fabian Hurzeler set out with a bold selection that included five forwards in his starting XI.
However, that backfired massively as the Seagulls were wide open and punished by Forest’s frightening pace and precision going forward.
Dunk gave the hosts a helping hand they did not need after just 12 minutes as he turned Gibbs-White’s dangerous low cross into his own net.
On his 100th Forest appearance, Gibbs-White then rose highest to power in a header from Anthony Elanga’s corner.
Manchester United will wonder about what they let go in Elanga, as the Swedish international cut Brighton open at will down Forest’s right side.
His pinpoint cross on the half-hour mark left Wood with the simple task to head in his 15th Premier League goal of the season and set his own personal best-ever tally for a season in the English top flight.
Two more were to come after the break for the New Zealander, who is enjoying the best spell of his career at 33.
Elanga was again the creator for the fourth as he dashed towards the byline and squared for Wood to tap in from point-blank range.
Gibbs-White was then manhandled to the floor by Tariq Lamptey from a corner, allowing Wood the chance to complete his hat-trick from the penalty spot.
Wood is now closing in on Stan Collymore’s record of 22 goals as Forest’s top scorer for a single season in the Premier League era.
Brighton remain in ninth after a damaging day for their own European ambitions.
After a bright start to Hurzeler’s reign, the 31-year-old has won just two of his last 11 league games.
It went from bad to worse for the visitors in the closing stages as Williams smashed in from close range.
Brighton goalkeeper Bart Verbruggen compounded a miserable day for his side in stoppage time as his pass was intercepted by Jota Silva, who surged forward and fired past the Dutch international.
