Sigurdsson fired into the roof of the net after Dave Hibbert had appeared to handle Ben Davies' cross after 22 minutes.
Davies added two more before half time - the first a fine 20-yard shot into the far top corner after good work by Kevin McIntyre in the 33rd minute and then bundled to ball over the line after Shwan Jalal had parried Grant Holt's drive.
Luke Daniels saved brilliantly from Michael Rankine and Shrewsbury lost their poise after the break.
Bournemouth pulled one back through substitute Brett Pitman after Michael Jackson had totally missed a header.
Joel Ward was dismissed in the 78th minute after two bookings inside 11 minutes and substitute Richard Walker added Shrewsbury's fourth in the third minute of injury time.