The striker scored with eight minutes remaining at the Liberty Stadium after a clever cutback from substitute Manasse Mampala.
Just a minute into the match, visiting goalkeeper Joao Virginia made a fine save to deny Liam Cullen.
Fraser Hornby also came close for Everton when his shot was deflected just over, before Broadhead's decisive strike.
With Leicester City and Derby County both losing, Everton went four points clear of second-place Brighton & Hove Albion on 31 points.
Swansea remain bottom with 11 points.