Burnley and Brighton enjoy impressive wins.
Ashley Barnes' double was key as Burnley scored three times in the space of 12 second-half minutes.
Crystal Palace and 10-man Everton played out a goalless draw, while Sheffield United grabbed a late equaliser to claim a point at Bournemouth.
MISERABLE STARTS FOR SAINTS AND WATFORD
Ashley Barnes' double was key as Burnley scored three times in the space of 12 second-half minutes to stun Southampton in a 3-0 win at Turf Moor, Johann Gudmundsson's impressive solo goal putting the icing on the cake.
New signing Neal Maupay, who arrived for a reported £20million, scored on his debut to help Brighton defeat Watford by the same scoreline at Vicarage Road.
The former Brentford striker followed fellow substitute Florin Andone by netting in the second half, Abdoulaye Doucoure's dreadful own goal after 28 minutes having set Graham Potter's new side on their way.
SHARP GRABS LATE POINT FOR BLADES
Substitute Billy Sharp finished off a goalmouth scramble with two minutes remaining to earn newly promoted Sheffield United a 1-1 draw away to Bournemouth.
Chris Mepham had earlier put the hosts in front when he converted a close-range rebound, but Sharp ensured the Blades got something from their first Premier League game since 2007.
There was also a draw as Crystal Palace held Everton at Selhurst Park, with the visitors having Morgan Schneiderlin sent off with 14 minutes remaining.
The midfielder picked up a second yellow card for a high foot Luka Milivojevic, while Wilfried Zaha - who reluctantly had to stay at Palace having been heavily pursued by Everton in the transfer window - played the last 25 minutes after coming off the bench.