(Villarreal win 3-1 on aggregate)
Sonny Anderson and Roger pounced on poor defending to end Celtic's dream of a second Uefa Cup final in a row.
Brazilian striker Anderson made Bobo Balde pay for early hesitancy to head Villarreal into the lead in the quarter-final second leg in Spain.
Stephen Pearson should have done better than head over for injury-hit Celtic.
But Roger nipped in ahead of an exhausted Joos Valgaeren to power home a volley from 16 yards to clinch a deserved victory for the Spanish side.
Celtic, last season's runners-up, had suffered a blow before kick-off with striker Chris Sutton failing a late fitness test.
Fellow Englishman Alan Thompson had already been ruled out through suspension, so Liam Miller came into midfield and Jackie McNamara into defence.
Just as in the first leg, Celtic conceded an early goal.
Juliano Belletti hung the ball in the air from the right and Anderson reacted quicker than Balde to head the ball over goalkeeper David Marshall after six minutes.
Jose Mari came close to making it 2-0, beating Balde for pace, but the former AC Milan striker stabbed his shot just wide from the edge of the box.
Juan Riquelme also drove across the face of the Celtic goal after Neil Lennon lost control at the edge of his own penalty area.
Celtic had claims for a penalty waved away by the referee after 30 minutes, when Rodolfo Arruabarrena challenged McNamara in the box.
Another thrusting run and superb cross from McNamara handed Pearson a great chance to equalise, but the midfielder headed powerfully over the bar from 12 yards.
Anderson missed a glorious chance of a second goal when he failed to connect with a cross with his head when only eight yards out.
A Henrik Larsson header across the face of goal just evaded Valgaeren as Celtic pressed after the break.
Balde produced a superb saving tackle as Anderson seemed poised to stroke home a killer second goal.
That finally came in the 68th minute, when Anderson crossed from the left of the box and Roger showed more desire than Valgaeren and blasted the ball past Marshall.
Villarreal: Reina, Arruabarrena, Coloccini, Belletti, Ballesteros, Josico, Roger, Battaglia, Riquelme, Jose Mari (Javi Venta 85), Anderson.
Subs not used: Lopez Vallejo, Marti, Victor, Guayre, Quique Alvarez, Calleja.
Booked: Josico, Battaglia, Arruabarrena, Javi Venta.
Celtic: Marshall, McNamara, Valgaeren, Balde, Varga, Agathe, Lennon, Miller (Smith 71), Petrov, Larsson, Pearson (Wallace 71).
Subs: Douglas, Sylla, Lambert, Wallace, Mjallby, Beattie, Smith.
Booked: Valgaeren, Pearson.
Referee: Massimo De Santis (Italy).