Walcott then turned provider as Richarlison doubled the lead, battling his marker and sending a header thumping past Alex McCarthy for his third Everton goal in two games.
Before the break, Jordan Pickford produced a remarkable sprawling save to deny Danny Ings when his own error had left the striker an open goal to aim at.
But the former Liverpool man did find the net nine minutes after half-time when turning home from a clever Mario Lemina flick at James Ward-Prowse’s corner.
Ings was totally unmarked in the six-yard box, and his goal opened a far more positive period for Southampton, but it was Walcott who should have sealed the points for Everton with 20 minutes to play.
The 29-year-old was played in by a delightful Gylfi Sigurdsson improvised flick, but fired horribly wide from inside the box.
Instead it was a slightly nervy end to a game that should have been far more comfortable Everton, with Pickford punching a fierce Ward-Prowse free-kick clear of his near post as Southampton pressed for a later equaliser.
Last week’s 2-2 draw away to Wolves already gave Everton fans a flavour of what to expect from Silva’s time in charge. There was much more adventure than was observed under Sam Allardyce – even playing with 10 men against a team that is expected to be competitive in England’s top flight.
At Goodison on Saturday, Everton once again impressed going forward, and especially impressive was Richarlison. The Brazilian was once again lively and full of tricks, linking well with those around him as Silva’s side conjured seven shots on target in all.
Perhaps most impressive however was the way in which Everton responded to Southampton dragging themselves back into contention – by continuing to attack.
Saints had come back into the game from a distant position after Ings’ goal, but the persistent quality of Everton’s play going forward ultimately saw them leave the pitch as deserved winners, applauded by their fans.
They should have added at least a third goal, but that reaction was a world away from the previous home match here – a 1-1 draw with a Southampton side battling for survival on 5 May, when there were boos at the final whistle.
That result sealed an eighth-placed finish for Allardyce’s team in the league. Can Everton hope for more this year?
A clue may lie in the several high-profile signings who are yet to feature this season. Yerry Mina has a minor foot injury, Andre Gomes is still recovering from a hamstring problem and Bernard is not quite match fit – so there may well be much better still to come.
BBC