A first top-flight win since a 3-1 victory at Bournemouth on December 18 provides Saints with a timely confidence boost ahead of Wednesday's EFL Cup semi-final second leg against Liverpool, but they may be without Van Dijk for their trip to Anfield.
The Dutchman hobbled off in the 56th minute after attempting to play through a knock sustained in the second period, potentially leaving Claude Puel short of options at centre-back in midweek with Jose Fonte having completed his £8million switch to West Ham on Friday.
James Ward-Prowse impressed as England manager Gareth Southgate watched on from the stands, opening the scoring with a sweet cross-goal strike from the right-hand side of the box in the 26th minute - his first goal in any competition since September.
Ward-Prowse then supplied the set-piece that led to Jay Rodriguez tucking home 13 minutes later, and Dusan Tadic wrapped up the points from the penalty spot after Wes Morgan felled substitute Shane Long.
Leicester have now conceded one more goal than they did in the entire of last season's title-winning campaign.
The Foxes also remain winless away from home in the top flight and are just five points clear of the drop zone following Swansea City's win at Liverpool on Saturday, while Saints are one point adrift of the top half.
Leicester reverted to a back four after deploying a three-man defence in a 3-0 loss to Chelsea last time out, but Saints width caused them problems with Kasper Schmeichel getting down to his left to stop Nathan Redmond converting from Tadic's deflected cross.
Schmeichel was beaten in the 26th minute, though. Ward-Prowse was left in space just inside the box and swept a fine right-footed finish from Cedric's cut-back inside the far post.
Oriol Romeu passed up a great chance to double Southampton's lead when he failed to connect with a corner while unmarked in the six-yard box, but it was 2-0 before half-time.
Under pressure from Maya Yoshida, Robert Huth could only nod Ward-Prowse's free-kick down to Rodriguez, who made no mistake with a clinical finish.
Van Dijk attempted to play on after sustaining an injury early in the second half but limped off and was replaced by Jack Stephens.
An errant pass from Christian Fuchs put Pierre-Emil Hojbjerg one-on-one with Schmeichel, only for the former Bayern Munich midfielder to miss the target again.
Morgan contrived to spoon over from inside the six-yard box after Huth cushioned Marc Albrighton's corner down for him, and the centre-back put through his own net in the 76th minute.
The Jamaica international headed Tadic's in-swinging free-kick past Schmeichel but was let off the hook by an extremely harsh, and potentially incorrect, offside call against Maya Yoshida.
Long replaced Rodriguez in the 80th minute and his pace proved too much for Morgan, who pushed him to ground inside the box to allow Tadic to round off the win by rifling home an emphatic penalty.