Captain Dawson put the Shakers in front after nine minutes when he collected Andy Morrell's throughball to slot past Shwan Jalal.
Bournemouth almost drew level when Bury goalkeeper Wayne Brown made an impressive fingertip save from Jeff Goulding's header.
Eight minutes before half-time, though, Bury were two up when Worrall cut inside Warren Cummings before beating Jalal at his near post.
Bournemouth substitute Steve Fletcher had a goal disallowed in the second half before Bury's Efe Sodje sliced against his own crossbar.
The Cherries did find the net in the final minute when Brett Pitman latched on to Fletcher's header to fire home but it proved mere consolation.