Odelusi, who has emerged from the youth ranks at the Reebok Stadium, helped himself to two well-taken goals either side of half time to kill off Shrewsbury's hopes of an upset.
Robert Hall had earlier driven Bolton in front from just outside the penalty area before Aaron Wildig hauled Shrewsbury level.
"When we first came to the football club Sanmi was a player that we felt had a lot more to offer," said manager Freedman. "I've got to give full credit to Jamie Fullarton who got hold of him and he's got our development squad working really hard.
"That's what I like to do when these cup matches come along. I don't want to go and buy anyone else. I want to look at my development squad and pick them.
Freedman made seven changes to his side which kicked off the season with a 1-1 draw at Burnley last weekend and was pleased with the contribution of the men drafted in.
"It was really a pleasing night for me because I can start to see what my team's all about," he added. "I'm happy with the way we approached it. That's the main thing."
Shrewsbury boss Graham Turner felt there were several positives from his side's display but was frustrated with how Bolton's goals went in.
"It was disappointing the way we conceded the goals," he admitted. "But I think it's encouraging the way we stuck at it and the way we played at times.
"We might have scored more, we got ourselves in good positions, but missed the target from several good shooting opportunities."