Rodriguez, playing in his preferred role as central striker, shone in a much-changed Saints side and scored 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."
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.
Wednesday have had their own injury problems to deal with but manager Dave Jones was more concerned with his side's defending on Tuesday night 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."
He added: "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."