Aljofree appeared to injure his neck in a clash with Rasiak after 21 minutes and there was a seven-minute delay as the Plymouth player was treated on the pitch by medical staff.
Aljofree was taken to hospital for treatment as Plymouth did well to cover his absence in a hard-fought match.
The referee added on an extra 14 minutes because of the incident and other injuries in the first half and there was another three minutes at the end of the game but Southampton were always marginally the better side and their victory lifts them up to fourth place in the Championship.
Plymouth were the brighter side for the first few minutes until Aljofree's injury forced a reorganisation.
Manager Ian Holloway sent on Marcel Seip to replace him and then made another change before the interval with Paul Wotton taking over from Lee Hodges.
Pilgrims goalkeeper Luke McCormick made a smart early save from the onrushing Nathan Dyer and, in the stoppage time added on, Kenwyne Jones should have put Southampton ahead but, with only the goalkeeper to beat, he allowed McCormick to rush out to save.
McCormick also did well to smother a fierce left-foot shot from Rudi Skacel as Southampton ended the elongated first half firmly in control.
Rasiak could have had a hat-trick in the second half with Seip blocking a shot of his after 49 minutes on the line.
But three minutes later the Polish international was on hand inside the six-yard box to push the ball over the line after Pele had seen his shot blocked.
Rasiak also hit the bar in the 72nd minute after substitute Bradley Wright-Phillips had teed up a shot.
With Plymouth committed to all-out attack Saints should have added another in injury time at the end of the second half but Rasiak allowed McCormick to delay him after being put clear by Wright-Phillips.
The referee had problems other than time-keeping, booking David Norris for failing to retreat at a free-kick and then showing a yellow card to John Viafara for a late foul.