The highly mutated and transmissible Omicron (BA.1) and its more contagious lineage BA.2 have provoked serious concerns over their decreased sensitivity to the current COVID-19 vaccines and evasion from most anti-SARS-CoV-2 neutralizing antibodies (NAbs). In this study, we explored the possibility of combating the Omicron and BA.2 by constructing bispecific antibodies based on non-Omicron NAbs. We engineered 10 IgG-like bispecific antibodies with non-Omicron NAbs named GW01, 16L9, 4L12, and REGN10987 by fusing the single-chain variable fragments (scFvs) of two antibodies through a linker and then connecting them to the Fc region of IgG1. Surprisingly, 8 out of 10 bispecific antibodies showed high binding affinities to the Omicron receptor-binding domain (RBD) and exhibited extreme breadth and potency against pseudotyped SARS-CoV-2 variants of concern (VOCs) including Omicron and BA.2, with geometric mean of 50% inhibitory concentration (GM IC) values ranging from 4.5 ng/mL to 103.94 ng/mL, as well as the authentic BA.1.1. Six bispecific antibodies containing the cross-NAb GW01 not only neutralized Omicron and BA.2, but also neutralized the sarbecoviruses including SARS-CoV and SARS-related coronaviruses (SARSr-CoVs) RS3367 and WIV1, with GM IC ranging from 11.6 ng/mL to 103.9 ng/mL. Mapping analyses of 42 spike (S) variant single mutants of Omicron and BA.2 elucidated that these bispecific antibodies accommodated the S371L/F mutations, which were resistant to most of the non-Omicron NAbs. A cryo-electron microscopy (cryo-EM) structure study of the representative bispecific antibody GW01-16L9 (FD01) in its native full-length IgG form in complex with the Omicron S trimer revealed 5 distinct trimers and one novel trimer dimer conformation. 16L9 scFv binds the receptor-binding motif (RBM), while GW01 scFv binds a epitope outside the RBM. Two scFvs of the bispecific antibody synergistically induced the RBD-down conformation into 3 RBD-up conformation, improved the affinity between IgG and the Omicron RBD, induced the formation of trimer dimer, and inhibited RBD binding to ACE2. The trimer dimer conformation might induce the aggregation of virions and contribute to the neutralization ability of FD01. These novel bispecific antibodies are strong candidates for the treatment and prevention of infection with the Omicron, BA.2, VOCs, and other sarbecoviruses. Engineering bispecific antibodies based on non-Omicron NAbs could turn the majority of NAbs into a powerful arsenal to aid the battle against the pandemic.