Jurgen Klinsmann's men came into Sunday's match having not tasted success since a 1-0 triumph over Czech Republic in September.
However, the USA got back to winning ways by prevailing in a physical encounter at the StubHub Center.
Bradley scored direct from a corner in the 27th minute to open the scoring in unconventional fashion, with Dempsey adding a second 10 minutes later.
The hosts began brightly, with Bradley involved in a number of early attacks that did not provide reward.
Jaime Penedo of Panama was then beaten in his 100th international appearance as a left-wing corner from Bradley drifted over the Los Angeles Galaxy goalkeeper's head and into the far corner without being touched.
Penedo was beaten again when Galaxy team-mate Gyasi Zardes cut in from the right to play in Dempsey.
The experienced Seattle Sounders attacker shimmied his way past the keeper before firing into the open net to double the USA's advantage.
Panama spurned a golden chance to pull a goal back when Matt Besler slipped while attempting to control the ball on the edge of his own area. Striker Blas Perez pounced, only to fire wide of Nick Rimando's goal.
A feisty first half contained 22 fouls and referee Henry Bejarano was forced to stop play repeatedly after the interval.
The USA were denied a third goal when Penedo saved a Jermaine Jones header from Bradley's neat free-kick, but there was little in the way of action thereafter as the tempo dropped amid a raft of substitutions.
Perry Kitchen and Matt Hedges made their debuts for the USA as replacements, with Miguel Ibarra and Zardes having been handed their first starts at international level.
