The stalemate at the Amex Stadium keeps Gus Poyet's men sixth in the table.
The Seagulls hit the woodwork twice during stoppage time but could not find their way past Chris Powell's hard-working side.
They could have climbed up to fifth with a victory, and thought they had secured one when Vicente rattled the crossbar and Gordon Greer hit a post.
Albion were at least able to maintain their unbeaten home record in 2013 but missed a big chance to open a gap on the chasing play-off pack.
Brighton dominated throughout with Wayne Bridge overlapping from left-back to provide an early chance for Leonardo Ulloa, whose shot was blocked by Michael Morrison.
Matt Upson the rose highest to meet Vicente's corner but his header was too close to Addicks keeper Ben Hamer.
Their best chance of the first half came when a short corner routine ended with Liam Bridcutt chipping the ball into the area and Inigo Calderon turning and firing inches wide from eight yards.
Yet Charlton boast one of the best away records in the Championship and it was soon clear why as they managed to stifle the hosts as the half wore on.
However, that meant they offered little threat in attack with their only goalscoring opportunity coming from Yann Kermorgant's shot which flew over the top.
The visitors fired another warning early in the second half when young winger Callum Harriott cut inside and drilled an angled shot just wide.
That stung Brighton into life, though, and they should have opened the scoring on the hour when Vicente got to the byline.
The former Spain midfielder's cross left Will Buckley with a free header but the winger nodded wastefully wide.
Dorian Dervite flashed a header wide for Charlton but Brighton remained in charge and Calderon fired off target again after the Addicks made a meal of David Lopez's corner.
Vicente's quick feet then gave him a glimpse of goal but his 20-yard drive was tipped wide by Hamer.
Dale Stephens brought a fine save out of Tomasz Kuszczak with a long-range effort before the assistant referee indicated four minutes of stoppage time.
It was the most dramatic four minutes of the match as Hamer tipped Vicente's shot onto the crossbar, and from the resulting corner Greer's header came back off the inside of the post.