An open first half saw plenty of chances at both ends. Craig Davies had already headed against the bar for the Championship club by the time Robert Hall drove Bolton in front with a low shot from the edge of the box in the 26th minute.
Shrewsbury were level within two minutes, though, with midfielder Aaron Wildig bursting into the penalty area to coolly convert an inviting right-wing cross from Paul Parry.
Wildig and Luke Summerfield then forced fine saves from goalkeeper Andy Lonergan before Bolton regained the lead two minutes from the interval.
Hall turned provider with a cross from the right which was steered high into the net by left winger Odelusi.
Bolton, showing seven changes from the side which started at Burnley last weekend, moved further ahead six minutes into the second half after catching Shrewsbury on the break.
David Ngog led a breathtaking counter attack before setting up Odelusi who drove home an unstoppable rising shot from the edge of the penalty area to crown a memorable night for the youngster.