Catalyzes the attachment of asparagine to tRNA(Asn) in a two-step reaction: asparagine is first activated by ATP to form Asn-AMP and then transferred to the acceptor end of tRNA(Asn). ATP + L-asparagine + tRNA(Asn) = AMP + diphosphate + H(+) + L-asparaginyl-tRNA(Asn) Belongs to the class-II aminoacyl-tRNA synthetase family.