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.
Shane Watson gave Rogers some valuable assistance, hitting 68 in a 129-run stand for the fifth wicket that helped the tourists wrest back command of a tight match after being reduced to 49-3 and then 76-4 in extremely tough batting conditions.
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.
Rogers spent an agonizing 19 balls on 96, playing and missing a couple of times and finding fielders with his shots, before sweeping offspinner Swann for four to finally get to three figures.
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.
Back from the wilderness
It's been a long road to becoming Australia's second oldest player to hit a first Test hundred, after the 37-year-old Arthur Richardson in 1926 at Headingley. 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.
That earned him a surprise call-up by Australia for this Ashes series. He scored 84 in the drawn third test at Old Trafford but has taken the next step here, underpinning the team's reply. It didn't come easy. Rarely will a century be so stop-and-start and he will have nightmares about some of Broad's deliveries to him across two devastating seven-over spells either side of lunch by the paceman.
Broad found pace, bounce and startling seam and swing movement, with some deliveries turning into quick leg breaks. His figures could have been much better and he beat the edge countless times, mainly that of Rogers.
Broad proved too much for David Warner (3), Usman Khawaja (0) and captain Michael Clarke (6) before lunch and should have had Rogers at the start of the second session, only for Swann to drop a diving chance at second slip. Rogers was seen laughing to himself at times after being befuddled so many times by the rampant Broad.
The earlier escape via DRS continued the theme of controversial umpiring that has blighted the series. Rogers was given out caught behind, but the ball had actually struck his pads and he went to DRS to correctly overturn the decision. However, the replays showed he could have been out lbw if England had appealed for that instead.
When Steve Smith (17) fell to Tim Bresnan nine balls into the second session, Australia was up against it but Rogers made hay with Watson after Broad went out of the attack.
Watson also survived a dropped catch when on 5, with Bresnan missing a one-handed caught and bowled, and he passed his first fifty of the series before tickling one of Broad's worst deliveries of the day down the legside for Prior to take a smart catch.
England earlier failed to add to its overnight total of 238-9, with Jackson Bird bowling James Anderson on the 12th delivery of the morning.