Lansbury levelled the scores just after half-time, with Forest going in at the interval trailing to Andrew Crofts' header just after the half hour mark.
Two goals in three minutes put Forest in the driving seat, with Lansbury slotting home from the spot almost immediately after Darius Henderson had given them the lead before Nathaniel Chalobah was sent off in the 67th minute as Forest finished the match with 10 men.
Brighton went into the game having scored only four goals in their previous five matches and, in a lively start, former Newcastle winger Kazenga LuaLua provided an immediate threat for the Seagulls.
Forest were chasing a club record-equalling seventh straight win but it looked like manager Billy Davies would taste defeat for the first time in eight games when Will Buckley put the visitors back in front with five minutes remaining.
Poyet though had sympathy for the goalkeeper and said: "To begin with I thought the shot must have taken a deflection but when I've seen it again I nearly killed myself!
When you are a keeper you pay the price and Casper has done that today. He was having a very, very good game, making two or three good saves, coming for crosses and kicking very well.
Billy Davies' unbeaten return as Nottingham Forest manager was preserved by an injury-time mistake from Brighton goalkeeper Casper Ankergren.
Forest were chasing a club record-equalling seventh straight win,
but it looked like Davies would taste defeat for the first time in eight games when Will Buckley put the visitors back in front with five minutes remaining.However, Ankergren, excellent until this point, allowed Henri Lansbury's shot to squirm beneath his body and their rivals shared the points.
Brighton took the lead through Leonardo Ulloa's 57th-minute strike before substitute Lewis McGugan sparked the late drama with Forest's first equaliser in the 82nd minute.
Davies' side stay fifth in the table but Albion leapfrog Leicester and move up to sixth place.