The Seagulls were on the front foot for much of the match with Magpies goalkeeper Nick Pope producing a brilliant display to deny the hosts.
Story of the match
Callum Wilson thought that he had given the visitors the lead after finding the back of the net, but the goal was ruled out for a questionable high foot.
Pope was called into action again, spilling Trossard's next chance and getting a hand to an angled strike by Solly March, but with the ball looking as if it were rolling in, Kieran Trippier made a vital clearance.
On 20 minutes, Trossard was able to evade Trippier, Pope denying him, the Belgian snatching at the second attempt and Danny Welbeck unable to control the rebound.
March then exchanged passes with Pascal Gross and his angled shot was spilled by Pope with Trippier again coming to the visitors' rescue with a second vital clearance.
As halftime approached, the Newcastle goalkeeper again frustrated the hosts, saving from Lallana, who shot on the turn.
Four minutes into the second half, Lallana was again denied by Pope, who made a brilliant, diving, one-handed save and followed by tipping over March's drive.
After March and Joelinton were booked, the Magpies then had their best spell of the match as Veltman was forced to clear away a Saint-Maximin cross with former Brighton defender Dan Burn meeting the resulting corner from Trippier, but heading woefully off-target.
Eddie Howe then introduced Sean Longstaff and Jacob Murphy and soon after Mitoma burst down the left resulting in a chance for Veltman that struck Burn and appeared to be going in, Fabian Schar with a timely clearance.
Mitoma again showed his class, picking out Gross, but his strike from six yards was pulled wide.
Chris Wood was thrown on for Newcastle, but he played little role in the remainder of the contest as the visitors had their goalkeeper to thank for earning a point.
Player of the match: Nick Pope
The former Burnley shot-stopper was chosen over Martin Dubravka and on a day when his defense constantly allowed Brighton time and space and the Newcastle attack created very few quality chances, he showed why Howe picked him, making seven crucial saves in a magnificent performance.