It took the visitors just four minutes to go in front with Bednarek reacting quickest in the box to head beyond Viktor Johansson.
Rotherham responded quite well to the early setback and carved out a decent opportunity when Peter Kioso crossed for Jordan Hugill to fire just over.
The visitors were just inches off doubling their advantage when Ryan Fraser took aim from 25 yards, only for his effort to cannon off the post.
Intricate play around the box then led to a great chance for Stuart Armstrong but he was denied by the sharpness of Johansson.
There was nothing the Sweden international goalkeeper could do about Southampton's second goal though, which arrived in the 38th minute.
It came from a lightning move with Fraser putting through by a chipped ball from Taylor Harwood-Bellis. Fraser then had the awareness to cleverly backheel the ball to Adam Armstrong who tucked his shot into the bottom corner.
It was almost 3-0 moments later with Will Smallbone denied by a strong hand from Johansson.
Gavin Bazunu was called into action at the other end with Seb Revan unleashing an effort after showing skill down the left.
The hosts threatened again early in the second half as they sought a way back into the game, Sam Clucas trying another spectacular volley to match his goal of the month contender in January, but this time it drifted off target.
Adam Armstrong had a sniff of a chance when he was picked out by a clever ball from Smallbone but his heavy first touch allowed Johansson to deny him.
Rotherham were doing the bulk of the pushing in the second half with the visitors content to sit on their lead.
The Millers had a good chance to get one back but Kioso could not pick out substitute Sam Nombe, who had made a good run to the front post.
Southampton brought on January capture David Brooks in the 74th minute and he almost made an instant impact only to see his effort, after cutting in from the right flank, whistle over the bar.
Rotherham came close to an injury-time consolation as Ollie Rathbone caught a volley perfectly, but Bazunu was equal to it to ensure a clean sheet.