In　solving　the　system　of　Stokes　and　Navier-Stokes　equations,　lack　of　mass　conservation　has　been　viewed　as　the　critical　drawback　of　the　finite　element　methods　based　on　the　least-squares　(LS)　principle.　Although　many　modifications　have　been　proposed,　there　is　a　need　for　a　global　approach　that　improves　both　mass　conservation　and　momentum　conservation.　The　key　to　such　a　global　method　is　to　control　local　conservation,　which　is　weaker　than　to　control　the　residual　everywhere.　Accordingly,　a　new　method　named　conservation-prioritized　Moment　Least-Squares　(CMLS)　is　developed.　The　CMLS　method　emerges　from　the　moment　conditions.　Among　these　moment　conditions,　the　zero-order　moment,　which　exactly　expresses　the　local　conservation　condition　on　the　element,　is　prioritized　over　the　others;　thereby,　good　local　conservation　can　be　achieved.　The　advantages　of　the　CMLS　method　over　the　LS　method　are　demonstrated　by　conservation　errors,　convergence　studies,　and　numerical　accuracy　in　nonlinear　Navier-Stokes　tests.　Besides,　the　CMLS　method　retains　the　merits　of　the　LS　method:　it　has　a　symmetric　positive-definite　global　matrix　and　the　same　interpolation　for　both　velocity　and　pressure.(c)　2022　Elsevier　B.V.　All　rights　reserved.