The Blue Square Bet Premier side equalised twice shortly after the interval, but ultimately the Owls' superior class and fitness won the day.
It was a relatively quiet first half. Gary Teale bagged Wednesday's opener in the 11th minute as he latched onto Clinton Morrison's low cross across the six-yard box.
Giles Coke saw a curling effort strike the crossbar in the 35th minute, but the match lit up just after the break thanks to an amazing 12-minute spell which brought six goals.
Southport got their dream leveller in the 52nd minute when ex-Liverpool trainee Paul Barratt tucked home, but Neil Mellor put Wednesday back in front a minute later with a close-range finish. Southport sub Matty McGinn rifled in dramatically with his first kick to make it 2-2, but then Wednesday cut loose.
Morrison scored two quick goals inside three minutes - the first into the top corner from close in and the second a well-taken drive into the bottom corner - before defender Tommy Spurr tapped home to make it 5-2 in the 64th minute. Wednesday substitute Paul Heffernan almost made it six with five minutes to go, but his header struck the crossbar.