Following a goalless first half, Burnley seemed on their way through to the second round as they led 3-1 with seven minutes of normal time remaining thanks to two penalties from England Under-21 international Rodriguez and a fine strike from Charlie Austin.
However, League Two Burton had other ideas and, having earlier made it 2-1 through Cleveland Taylor, they sent the game to extra-time following late goals from Calvin Zola and Jacques Maghoma.
Burton's hopes of an upset lasted barely three minutes of the additional time, though, as Ross Wallace and Rodriguez both netted quickfire goals from distance to put Burnley 5-3 ahead, and this time there was no way back for the visitors, with Rodriguez adding another in the 106th minute to seal the win.
There was little sign of the drama to come as the first half produced few clear-cut chances for either side, but the deluge of goals began when Rodriguez put npower Championship outfit Burnley ahead from the spot in the 57th minute after Austin was fouled by Tony James.
Nine minutes later it was 2-0 with Rodriguez once again making no mistake from 12 yards out, this time after Wallace had been impeded by Adam Bolder, but Burton halved the deficit through Taylor in the 72nd minute.
Burnley restored their two-goal advantage through Austin's thumping 83rd-minute shot into the top corner, but Eddie Howe's men were stunned as Burton hit back again with goals from Zola and Maghoma, the latter in stoppage time.
That sent the contest to extra-time, but any hopes Burton had of sealing a memorable win were crushed almost immediately as Burnley netted two superb long-range goals.
Wallace made it 4-3 in the first 60 seconds with a thumping 25-yard drive and then two minutes later Rodriguez made it 5-3, firing in a superb strike from 30 yards out to compete his first senior treble.
On what was an unforgettable night for the home-grown 22-year-old, Rodriguez added his fourth of the night to complete the scoring following a one-two with Wade Elliott.