Centre-half Van Dijk, the £13million deadline-day signing from Celtic, headed an early opener before Tadic struck at the start of the second half.
Mane then added the third to make it nine goals for free-scoring Saints this week following their Capital One Cup demolition of MK Dons.
Swansea, by contrast, are now without a win from their last four matches, although they at least registered their first goal in that stretch when Gylfi Sigurdsson tucked away a penalty.
However, the Swans were well beaten by then, as Saturday’s first England-versus-Wales showdown went to the home side.
The Swans could have taken the lead in the sixth-minute when Maarten Stekelenburg parried Jonjo Shelvey’s shot into the path of Ki Sung-Yueng.
But unfortunately for Ki, the South Korean midfielder’s finish would have looked more at home at Twickenham as he blazed high over the crossbar.
And the opening goal arrived in the 10th minute when Van Dijk shook off the attentions of his marker, former Saint Jack Cork, to power in a near-post header from James Ward-Prowse’s corner.
Swansea briefly threatened to rally before half-time but Shelvey’s shot flew over and Stekelenburg saved well low down from Ki.
Garry Monk made a double change at half-time, hauling off Cork and the ineffective Bafetimbi Gomis.
But Southampton gave themselves some breathing space in the 54th minute when Pelle’s neat chip sent Tadic scampering in behind.
The Serbian midfielder looked to have gone too wide but as he cut back inside and shot, the ball looped up off the boot of Ki and flew over Swansea goalkeeper Lukasz Fabianski.
And just as Saints fans began regaling their Welsh visitors with a chorus of Swing Low Sweet Chariot, Mane popped up with his fourth goal of the season.
Ashley Williams only half-cleared after Pelle prodded the ball goalwards and Mane was on hand to guide a precise, low shot inside Fabianski’s right-hand post from 20 yards.
Swansea did at least manage a consolation, ending their recent goal drought, after Neil Taylor was brought down by Jose Fonte in the 82nd minute and Sigurdsson stepped up to tuck in the spot-kick.