Building upon our arithmetic duality theorems for $1$ -motives, we prove that the Manin obstruction related to a finite subquotient ${\cyrille B} (X)$ of the Brauer group is the only obstruction to the Hasse principle for rational points on torsors under semiabelian varieties over a number field, assuming the finiteness of the Tate-Shafarevich group of the abelian quotient. This theorem answers a question by Skorobogatov in the semiabelian case and is a key ingredient of recent work on the elementary obstruction for homogeneous spaces over number fields. We also establish a Cassels-Tate-type dual exact sequence for $1$ -motives and give an application to weak approximation