Quashie was making only his fourth start in a Southampton shirt since completing his £2miilion move from arch-rivals Portsmouth in the January transfer window and provided the game's telling moment five minutes after the interval.
The goal was a triumph for Southampton manager Harry Redknapp who had introduced Swede Mikael Nilsson as a substitute for the injured Rory Delap just seconds earlier.
Nilsson delivered an inviting inswinging centre from the right which Quashie had no trouble in heading home past stranded goalkeeper Paul Robinson from six yards out.
The win was only Southampton's second in the league since Redknapp took charge in December and improved their chances of moving out of the relegation zone.
But Southampton's hero for the second game running was goalkeeper Paul Smith who produced another outstanding display as he continues to fill in for injured first-choice star Antti Niemi.
Smith produced save after save and at times formed a one-man barrier between his side and their Tottenham opponents, especially from England striker Jermain Defoe who had scored a hat-trick in the two sides' previous meeting.
Tottenham striker Mido should have opened the scoring for the visitors inside the opening minutes. Michael Brown dispossessed Jamie Redknapp on the edge of the penalty area before feeding the Egyptian but his low shot was saved brilliantly by Smith with his legs.
Senegal international Henri Camara has been a real threat to teams' defences since joining Southampton five weeks ago and his presence caused considerable confusion in Tottenham's defence on 15 minutes.
Strike partner Peter Crouch narrowly failed to connect with his cross from the left and, when the ball was retrieved, Camara then saw his shot charged down by Ledley King with the goal beckoning.
But Defoe should have finally provided the breakthrough eight minutes later after finding himself clear on goal with just Smith to beat.
As Southampton complained with injured defender Claus Lundekvam lying prostrate in Tottenham's penalty area, Defoe closed in on his 21st goal of the season but Smith somehow deflected his effort over the crossbar for a corner.
Smith's opposite number Robinson did well to block Camara's attempted lob on 32 minutes but he was grateful for a wasteful piece of finishing by Crouch in the dying seconds of the first half.
Camara raced onto Olivier Bernard's ball over the top to cross invitingly for his strike partner but the giant forward put his effort into the side-netting at full stretch.
Quashie gave the Saints the lead five minutes later and Robinson produced a wonder save to stop Crouch extending the lead on 53 minutes but it was Smith who shone as the star of the second half.
He somehow deflected Defoe's fierce right-foot shot for a corner within a minute of Saints taking the lead while he denied the same player from close range with a full-length stop four minutes later.
He finally looked as if he might be beaten with just three minutes remaining as substitute Fredi Kanoute burst through on goal but Smith once again came to the rescue to save his low shot with his legs.
MAN OF THE MATCH: Paul Smith (Southampton) - Was the Saints' saviour for a second successive match with a string of brilliant stops.