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.