Bad light brought an early end to an attritional day's play at Chester-le-Street, but not before Rogers had progressed to 101 not out and Australia had moved to 222-5, just 16 runs behind England's first-innings total.
Broad delivered a devastating display of pace bowling in overcast conditions, ripping through the top order to claim figures of 4-48.
But try as he might, he couldn't snare Rogers despite another controversial DRS call when the opener was on 20 and a dropped catch by Graeme Swann on 49.
Brad Haddin was at the crease on 12 with Rogers before umpires called the players in for bad light, with 38 minutes still possible in the day. The ground in County Durham, hosting its first Ashes test, has no floodlights.
England is 2-0 ahead in the series and has retained the Ashes. The Australians can still draw 2-2, though, which would give them a boost ahead of the return series Down Under starting November.
It seemed to take him a while to really grasp the size of his achievement. When he did, he slowly took his helmet off and raised his bat to the crowd, with the Australian balcony celebrating. In radio interviews after stumps, Rogers was close to tears.
"It is the sweetest moment of my cricket career," Rogers said. "After all this time, to play and get a test hundred is very satisfying."
Now 35, Rogers was in the test wilderness after being dropped after just one test, back in 2008 against India in Perth, and spent the next five years playing his trade and plundering plenty of runs on the county circuit in England and in Australian state cricket.
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