The 23-year-old has had limited opportunities since his £6million summer move from Burnley, with manager Nigel Adkins using him out wide when he has featured.
Against the Championship side, Rodriguez was played in his preferred role as central striker and shone in a much-changed Saints side, scoring his first goals for the club in their 2-0 defeat of Wednesday.
"He had a massive impact," assistant Crosby said. "I thought his movement was excellent against two centre-halves that perhaps wanted him to come short.
"I thought his movement in between and down the side of them was really good.
"He is one of many players who have taken their opportunity.
"He has had to bide his time and wait for his chance but he has been a great character around the place and wants to improve.
"He is at a good age for the club and has taken his opportunity.
"We know he is most comfortable in a central role but he has good competition in front of him at the moment.
"Whenever you get an opportunity, you've got to take it and he did that tonight. Whenever he is needed, we know he is more than ready."
Rodriguez's first goal came on the half-hour mark when he expertly turned his markers to rifle home past Stephen Bywater.
The Wednesday goalkeeper did manage to get to his penalty 12 minutes from time, but it trickled home to ensure Southampton's progress to the last 16.
"I think the way we set about the game made it a comfortable night," Crosby said.
"I think we were excellent first half, moved the ball really well and applied pressure to them high up the pitch.
"We controlled long periods of that game. At half-time the main thing was could we improve our quality in the final third. It was a great night's work."
The only negative for Southampton was a late knock picked up by Jack Cork on his first match back from an ankle injury sustained in pre-season.
"We don't know the extent of it at the moment," Crosby said.
"His studs got stuck in the turf a little bit so he has come off so we will assess that."
Crosby also revealed that several players have minor injuries that prevented them from taking part on Tuesday evening.
"Luke Shaw has been ill and we can't risk him on the back of that," he said.
"Tadanari Lee has unfortunately got an elbow injury, which may keep him out for a week or so.
"Emmanuel Mayuka when he nearly scored against Villa on Saturday hurt his ankle a little bit but nothing major. They will all be back in the next few days."
Wednesday have had their own injury problems to deal with but manager Dave Jones was more concerned with his side's defending tonight and in recent matches.
"We had chances but we have to defend," the former Southampton boss said. "We can't keep giving the goals away that we are.
"They are stoppable and you need a base to work from but at the moment we are just finding it, for whatever reason, a little bit hard to keep a clean sheet.
"We had a couple of good chances in the first half and again in the second half, but you can't give away the goals we are week in, week out and then say it is okay because it is not.
"There are some good players in there but at the moment maybe one or two are not doing what they should be doing."