Winger Hakim Ziyech's opportunist strike gave Thomas Tuchel's side the lead against the run of play in the 28th minute.
Chelsea began to dominate late on but could not force a winner as Brighton dug deep to preserve their point.
Chelsea's last win in the Premier League came in a 3-1 victory over Aston Villa on Boxing Day, since gaining three points from the last 12 available.
Brighton, who are in ninth place, have now drawn 11 games this season, and Webster said they were disappointed not to get all three points against the European champions.
"I think for the whole game we were the better team. They had a spell at the end but other than that we were the dominant team and on another day hopefully get three points," he said.
For all Brighton's bright approach play early on they struggled to land a telling blow and were stunned when Ziyech took aim from 25 yards and rifled a low shot that keeper Robert Sanchez could have done better with.
Brighton began the second half on the front foot and Danny Welbeck was played in by Pascal Gross but the striker got his angles wrong and blazed wide.
From the resulting corner Webster thundered a header into the net after finding space in the area.
"We looked tired," Tuchel said. "We tried everything but you could see we are mentally and physically tired."