Akinfenwa gave Northampton a sixth-minute lead, converting Marshall's low cross before having to leave the field injured.
Marshall and substitute Billie McKay went close to increasing Northampton's lead while Steve Guinan forced a save from Shwan Jalal.
But Brett Pitman squandered Bournemouth's best chances either side of the break. Pitman allowed Chris Dunn a save on both occasions when he should have done better.
Pitman missed another good chance when Dunn pulled off a point-blank save while at the other end McKay shot over following a powerful run from the impressive Marshall.
Substitute Alan Connell wasted two late chances for the Cherries while Jalal saved well from Andy Holt.
But Northampton finally grabbed a second decisive goal in the 85th minute when Marshall knocked home the rebound after Jalal had saved well from McKay.