Officers for the Lee County Sheriff's Department surrounded a Fort Myers Bank of America after a 36-year-old man attempted to rob the establishment in February. The culprit took 2 hostages inside the bank ... using them as shields while communicating with a negotiator.

The Sheriff's Department released video of the standoff -- with a sniper steadying his aim onto the suspect while right outside ... balancing his gun on another deputy's shoulder.

Despite the negotiator's attempts to talk down the robber -- he's heard referencing the man's children at one point -- the offender refuses to surrender ... and instead puts a knife to the throat of one of the hostages.

This prompts an immediate response from the sniper ... who takes a shot, with the bullet traveling through a desktop computer screen and right in between the 2 hostages -- successfully striking the robber.

As you can see in the footage ... the suspect falls to the floor as the hostages run off. Police then descend upon the scene, throwing in flash-bang grenades and doing a sweep.

Lt. Todd Olmer of the Lee County Sheriff’s Office added ... "Shooting positions like these are trained regularly by snipers for time-critical incidents like these. Due to the suspect’s limited exposure, the sniper took a planned and deliberate shot through a computer monitor, striking the suspect in the center of the forehead, killing him instantly."

As the Lee County Sheriff's Department later noted ... the hostages' safety was their top priority -- and fortunately, they achieved just that. Of course, it was a pretty close call.