A method and system for attaching a TLP to its tendons using pull-down lines to rapidly submerge the hull to installation draft while compensating for inherent hull instability during submergence and to provide motion arrest and aid in station keeping. The system includes tensioning devices mounted on the TLP, usually one for each tendon. Each tensioning device is equipped with a pull-down line which is connected to the corresponding tendon. The TLP hull is submerged to lock-off draft by applying tensions to the pull-down lines connected to the top of the tensions, or by a combination of applying tensions to the pull-down lines and ballasting the hull. As the tensioners take in pull-down line, the hull submerges, i.e. the draft increases. After lock-off, high levels of tension in the pull-down lines can be rapidly transferred to the connection sleeves by slacking the pull-down lines, thus allowing the TLP to be made storm-safe much faster than by prior art methods which require de-ballasting to tension the tendons. In concert with TLP installation, the method may be used attach the mooring tendons to the seabed by suspending and lowering the tendons into their foundation receptacle in the seabed.