The game got off to a flying start as Brighton pinned back the visitors for the first 10 minutes before Newcastle started to hit back with lightning-quick counter attacks.
Solly March looked to have given Brighton the lead in the 33rd minute but fullback Kieran Trippier was on hand to clear the ball off the line, with replays showing that most of the ball was over the line.
Tempers boiled over in the 57th minute though, when Joelinton and March got involved in a shoving match down by the corner flag, with both players getting shown the yellow card.
The water break shortly afterwards -- introduced this weekend due to scorching temperatures -- was well-earned following a goalmouth scramble that saw Brighton go close, only for Newcastle to zip up the other end on the counter in a move stopped by a last-ditch tackle on Miguel Almiron.
Newcastle continued to threaten on the break and from set pieces, but it was Brighton who went closest to breaking the deadlock, forcing a number of saves from Pope and a goal-line clearance from Fabian Schar as the game finished scoreless.
After winning their opening league fixtures last weekend, the draw leaves both sides on four points.
"The performance was fantastic. Although we should have won we're incredibly proud. We were playing against a top team," Brighton manager Graham Potter said.
"Credit to the goalkeeper, we just missed that last little bit. We knew how tough it was going to be against Newcastle, with the way they set up and how good they are on the counter-attack," he added.