Striker Romelu Lukaku opened the scoring on 28 minutes with a powerful header from a corner, but Welbeck came off the bench late on and scored to rescue a point for manager Graham Potter's side.
The draw pushed Chelsea into second place in the Premier League table on 42 points, one more than Liverpool who have a game in hand and eight points behind leaders Manchester City. Chelsea are due to play both Liverpool and Man City in the first two weeks of January.
Brighton sit in 10th place on 24 points.
Lukaku, who scored for the first time in September in a 3-1 win over Aston Villa on Sunday, nodded his side into the initial lead when the 28-year-old connected with midfielder Mason Mount's corner at the near post, powerfully sending it past opposing goalkeeper Robert Sanchez. The Video Assistant Referee (VAR) conducted a lengthy review into whether Lukaku's trailing arm fouled Brighton striker Neal Maupay as he moved toward the ball, but the goal was awarded.
The home side claimed for a penalty shortly after half-time when winger Christian Pulisic fumbled over the ball but not before being taken down in the box, although the claims were dismissed after a VAR review.
Chelsea's Callum Hudson-Odoi squandered a glorious chance to double his side's lead on 55 minutes when he raced away on the counter attack with only one defender and the goalkeeper to beat, but the 21-year-old failed to find Mount with his final pass.
Brighton nearly made Chelsea pay moments later when midfielder Alexis Mac Allister collected a through ball and took the ball wide around goalkeeper Edouard Mendy, but Maupay could not strike home when set up for the finish.
Brighton continued to apply pressure, and they were rewarded in stoppage time as Welbeck headed home from a corner. The goal was the first the club had ever scored at Stamford Bridge.
"We have simply way too many players who we don't know what we can demand," Chelsea manager Thomas Tuchel told reporters after seeing his side held to a 1-1 draw for the fourth time in five home league games.
"We have seven COVID cases, we have four or five players out for six weeks," the German said when asked about Chelsea' chances of winning the league. "Why should we compete in a title race?"
James suffered a hamstring injury while Christensen went off with a back problem, he said.
Potter praised his Brighton players for pushing right to the end to get the point.
"I thought it was a fantastic performance from the players really full of courage full of quality," Potter said.