The　inhibition　phenomenon　of　Rayleigh-Taylor　instability　by　surface　tension　in　stratified　incompressible　viscous　fluids　driven　by　gravity　has　been　established　in　[Y.J.　Wang,　I.　Tice　and　C.　Kim,　Arch.　Ration.　Mech.　Anal.,　212　(2014),　pp.　1-92]　via　a　special　flattening　coordinate　transformation.　However,　it　remains　an　open　problem　whether　this　inhibition　phenomenon　can　be　rigorously　verified　in　Lagrangian　coordinates　due　to　the　delicate　nonlinear　part　of　the　surface　tension　term.　In　this　paper,　we　provide　a　new　mathematical　approach,　together　with　some　key　observations,　to　prove　that　the　Rayleigh-Taylor　problem　in　Lagrangian　coordinates　admits　a　unique　global　-intime　solution　under　the　sharp　stability　condition　nu　＞　nu(C),　where　nu　and　nu(C)　are　the　surface　tension　coefficient　and　the　threshold　of　the　surface　tension　coefficient,　respectively.　Furthermore,　the　solution　decays　exponentially　in　time　to　the　equilibrium.　Our　result　provides　a　rigorous　proof　of　the　inhibition　phenomenon　of　Rayleigh-Taylor　instability　by　surface　tension　under　Lagrangian　coordinates.