Weiss' second goal, 12 minutes into the second period, put Florida ahead 5-1 but the Panthers had to weather a determined comeback in which the Capitals pulled to within a goal with 1:44 remaining in regulation.
Scott Clemmensen closed the door the rest of the way en route to a 28-save performance for the Southeast Division-leading Panthers, who bested the Caps for only the second time in five meetings.
Dmitry Kulikov added a goal and two assists and Brian Campbell contributed three helpers in the victory.
Washington dropped its fifth straight road contest.
Michal Neuvirth allowed five goals on 30 shots for Washington.
Celebrity Families of 2014 [PHOTOS]