The empire was definitely well spread out enough that this could have had a decent impact in a couple aspects...
The Western Roman Empire that broke off from the Eastern Empire was pretty well doomed and this invention wouldn't have saved them. That said...it would have helped due to their setup. The Roman army was divided into two types of forces. The first was the Limitanei. These were Roman citizens that lived in Frontier towns for their lives. The would frequently marry women from these frontier towns and become part of the town and people...raising their families et al. They weren't exactly the best equipped, but they could defend their town as required. Once a frontier town was attacked, the Roman garrison would send word to the second line of Roman defence. The second line of defence were Comitatenses...this was the professional Roman army that was designed for quick deployment using their well created road systems. The Limitanei would hold their ground defending a towns walls while they waited for the Comitatenses to ride in and break the seige. Quite effective in that one army could effectively garrison multiple frontier towns.
The impact in this sense would be extremely beneficial...the time that it took a Limitanei force to report the attack on them and the Comitatenses to start on their way would be minimal compared to what it took to dispatch a messenger instead...it would really make this tactic that much more effective.
Would it have stopped the Barbarian invasion? Unfortunately it's highly doubtful, the empire was heavily dependent on disappearing farming income and was pretty close to completely broke. Facing this broke military was entire tribes of people that uprooted themselves and marched their entire towns into Roman territory, in some cases sacking Rome a long the way. The Vandals marched through Europe and into spain, across into Africa, before settling into former Carthage (and sacked Rome from the south before being defeated by Byzantine). The Visigoths took up shop in italy after sacking Rome...the Franks moved into whats now France...the Lombardi people moved into Milan...the Huns came a long with. It's a long list, the Romans, even with near instant communication, couldn't resist this onslaught. If they had held onto France and the former Celtic territories, odds were they'd be facing a viking invasion as well. Roman power struggles didn't help much...an Eastern empire under Constantine started making deals where he just gave Western empire land away to the new tribes (franks in particular) and the will to maintain a Western empire pretty much disappeared.
The Western Roman Empire was cash strapped...and unfortunately this setup would cost a significant amount of money to build and maintain. Nearing the end of their demise, they were actively disbanding troops they couldn't afford to pay.
The Eastern empire would have had a decent go with the technology...messages from Constantinople over to Egypt would be quite quick. I'm not sure how this would translate into economic advantages though.