In Roman times there were no Anglo-Saxons. Angles were a specific ethinic group in Denmark & Germany, Saxons were another specific ethnic group in Northern Germany, and the Romans customarily referred to most tribes and ethnic groups in northern Germany as Saxons.
So the late Romans fought against many Germanic raiders in Gaul and Britain they called Saxons. And they may have hired many Germanic warriors in Gaul or Britain who they called Saxons. Some of those Saxons may have been ethnic Saxons, some may have been ethnic Angles, some ethnic Frisians, some ethnic Jutes, etc., etc., but the Romans called them all Saxons, when they didn't just call them Germans or barbarians.
But as far as I know nobody ever called himself or anyone else an Anglo-Saxon for five hundred years after that.
According to history or legends told centuries later by Bede, by the Historia Brittonum, and by the Anglo Saxon Chronicle, after the last Roman withdrawal about 410 the Britons hired "Saxon" mercenaries to defend them and eventually the "Saxon" mercenaries revolted and over decades more "Saxon" settlers, warriors, and kings moved to Britain and took over all of southern England before the mission of Saint Augustine in 597.
According to those stories the first Germanic people settled in the island of Britain were three shiploads of warriors sometime after the Roman withdrawal.
Even though there were enough Germanic mercenaries in the Roman Army in Britain in 306 AD for King Crocus of the Alemmani to be the Roman officer who proclaimed Constantine I Emperor at York.