Mark Eiglarsh says the decision came down from Broward County prosecutors on Wednesday ... after he tells us he successfully showed the allegations against Taylor stemmed from nothing more than the former football player's estranged wife making claims in order to "gain leverage in their bitter, three-year divorce battle."

"We are deeply grateful for the prosecutors in this case," Eiglarsh said, "ensuring that true justice was served."

Eiglarsh and Lawrence posed for a picture following the news ... with each sporting a big smile on their faces.

We've reached out to prosecutors for comment, although so far, no word back yet.

Taylor was initially arrested in the case in mid-July ... after officers at the Pembroke Pines Police Department claimed he had not lived at his registered address in several months. As a registered sex offender, authorities argued he should face a felony charge for failing to report his residence change.