Manager Paul Fairclough has decided to take a director of football role following a disastrous winter run, but Barnet, despite little possession, were clinical in front of goal.
In an opening period of few chances, they took the lead when the lively Adomah seized upon a long clearance to turn past two Bournemouth defenders and plant his finish beyond Shwan Jalal.
The Cherries attempted to stage a mini-revival as first Scott Guyett's dangerous cross was almost headed past his own goalkeeper by Barnet defender Michael Townsend.
The home side then struck the woodwork moments before the break as Barnet goalkeeper Ran Kadoch could only punch a Sammy Igoe cross as far as Danny Hollands whose half-volley clipped the top of the crossbar.
In the 70th minute, Abu Ogogo won a strong tackle in midfield to release Adomah once more and he rounded Jalal to claim his second of the game.