The Chronica Gallica, about the closest contemporary source being within about a decade, strongly suggests Britain and several other Roman territories were given over to and deliberately settled by the Anglo-Saxons, under the Roman authorities, rather than being completely abandoned as the Empire was in the process of withdrawing.It was claimed several centuries later that some Germanic mercenaries (Hengist and Horsa) were invited over for that purpose and paid to fight against the Picts...and then turned on their employer and took almost all of Britannia because they could.
Subsequent works also speak of Saxons being brought in and set up with land in key areas, precisely for the defence of the Britons. The Saxons were one of the Foederati, as the Romans called them, ie a bunch of fighty types with whom they had agreements and gave benefits (usually settlements within the Empire) for their military service.
Bede paints quite the picture of the Anglo-Saxons, almost as the God-given saviours of the filthy heathen Brits... or even their divine punishment/retribution.
Once established, a great many more Saxons hopped over to join the settlements and thus began the peaceful migration.
The strongly supported suggestion that the land was given to them anyway gives them no reason to take it... because they already have it.The very fact that the Romano-Britons needed mercenaries to protect them proved that they couldn't protect themselves, so why not take it all?
There was meant to be something in the foedus treaty about Rome sending food supplies, which the Saxons didn't get and so there was a bit of a kick-off over that which culminated in the Battle of Badon Hill and gives rise to various King Arthur stories. Nothing quite the same as a military invasion, though. More like a change of management after a company is sold off.