Quashie shot home from 12 yards after Norwich captain Craig Fleming handled following a flick-on from Kenwyne Jones.
The former Portsmouth midfielder kept his cool when forced to re-take his spot-kick after several Norwich players encroached.
Norwich though missed a host of chances, in particular Dean Ashton who found himself clean through after Darren Powell's slip.
The home side started the brighter with their midfield quartet of David Prutton, Quashie, Matty Oakley and Djamel Belmadi dictating the pace of the game.
Prutton went close on four minutes when he burst through the centre of midfield only to scuff a weak left-footed shot straight at Robert Green.
Saints though pressed and were rewarded for their dominance with Quashie's penalty.
However, on taking the lead, the home side then seemed to retreat into a defensive shell and allowed Norwich back into the game.
Darren Huckerby in particular caused problems for both Southampton full-backs with some probing runs.
And Norwich had their own penalty shout on 45 minutes when Huckerby was bundled over by Prutton close to the touchline but no Norwich player seemed to appeal so referee Lee Probert waved play on.
Saints' second-half did not get off to a good start when Claus Lundekvam was stretchered off with a head injury and that signalled further Norwich dominance.
Jim Brennan headed against the post from a Dean Marney cross, Youssef Safri hit a dipping shot from 30 yards which beat Antti Niemi but sailed over the crossbar and Brennan again came close when released by Ashton.
Manager Nigel Worthington threw on substitute strikers Leon McKenzie and Peter Thorne to bolster his attacking options but Saints held on to pick up their first home win of the season.