The former Burnley striker has been used sparingly by Saints' boss Nigel Adkins in the opening few weeks of the campaign after his £6 million summer move, with most of his match action coming on the left of a front three, but he netted both goals on Tuesday night in his first start at centre forward.
"It was really good for us to get through to the next round and put on a good show for the fans," Rodriguez, who scored 21 goals in all competitions for the Clarets last season, told Southampton's official website.
|10/1||Jay Rodriguez is 10/1 with William Hill to score the last goal in Everton v Southampton
"It was nice to get my first goal for the club. It's a nice feeling to score, especially at St Mary's. I love playing up front. I felt really good playing that role and getting in behind and trying cause the defenders as many problems as I can.
"It is a good feeling to play up front and that is where I like it. It is where I've grown up and learned to play but I will work as hard as I can for the team and play in any position.
"I will try my hardest in any position but, yeah, I do enjoy playing up front."
Rodriguez is aiming for the Saints to push on in the League Cup, adding: "I think we'll take any team that comes.
"Reaching the last 16 is a great achievement but we still want to go further, so we'll take any team and show them how good we are so we can hopefully get through to the next stage.
"It was really good for us to get through to the next round and put on a good show for the fans – it builds confidence."