The Saints - showing six changes from their memorable 2-1 FA Cup quarter-final win over Arsenal three days earlier - showed no signs of a hangover early on in north Wales.
They were twice thwarted by the framework of the Wrexham goal in a matter of seconds as Larin headed Ryan Manning's delivery on to the crossbar, with Azaz nodding the rebound against the bar.
But they were not to be denied four minutes later as Azaz picked out Matsuki, who drilled into the bottom corner to beat Arthur Okonkwo at the goalkeeper's near post.
Saints' attacking intent was rewarded again just 10 minutes later as James Bree's free-kick deflected off George Dobson and looped into the path of Downes, whose sweetly-struck volley from 15 yards out flew into the net.
The hosts looked shell-shocked as the travelling Southampton supporters revelled in their side's dominance.
Wrexham finally showed signs of life in attack after the half-hour mark. They almost reduced the Saints' lead when Max Cleworth's cross found Nathan Broadhead, whose glancing header struck the post.
They did pull a goal back a minute later as Okonkwo's long punt upfield was flicked on by Kieffer Moore, and Windass showed great composure to slot beyond Daniel Peretz.
The second half was a far more even affair - with Okonkwo doing enough to keep out Larin's shot, before Cleworth's deflected strike sailed wide at the other end.
But just as Wrexham's momentum was growing, Larin intercepted Lewis O'Brien's pass before charging down on goal and firing beyond Okonkwo. The strike was Larin's fifth goal since joining on loan from Mallorca on transfer deadline day in February.
The hosts refused to let their heads drop and Moore headed Issa Kabore's cross on to the base of the post from close range soon after.
The visitors managed to take the sting out of the contest though, and after Azaz rifled against the post, Stewart headed fellow substitute Welington's free-kick into the net.
Just two minutes later, Azaz capitalised on a defensive mix-up to put the gloss on his side's stellar showing, as Tonda Eckert's men stretched their unbeaten league run to 13 league games in emphatic fashion.