Hearts, fresh from a 6-1 League Cup win over Kilmarnock midweek attracted a crowd of 14,548 to Tynecastle for this match against United.
Hearts began where they left off in midweek, with Gary Mackay and Dave McPherson coming close with early efforts.
But Hearts goalkeeper Henry Smith had to be alert to palm away a Kevin Gallacher header.
Hearts’ Sandy Clark also had a header brilliantly saved before John Robertson opened the scoring on 21 minutes from the penalty spot after John Colquhoun was fouled by Maurice Malpas. Roberston was clinical from 12 yards.
Fifteen minutes later, Clark powerfully headed home an Ian Jardine corner kick. Colquhoun claimed he had got a touch on the ball but the goal was credited to Clark and the points appeared safe. Hearts remained well on top in the second half and, within five minutes came a third goal when Ian Jardine scored past United goalkeeper Thomson with an angled shot following good build-up play between Robertson and Clark.
A header from Gallacher caused little concern to Smith and, at the other end, a Colquhoun shot that hit the crossbar almost made it four, but on 77 minutes Clark pushed home a Colquhoun knockdown from a Sandy Jardine lob.
Iain Ferguson, later to join Hearts, scored a consolation goal for United, driving home a Maurice Malpas corner kick three minutes from time.
Hearts skipper Walter Kidd was booked in the closing stages.
Hearts: Smith, Kidd, Whittaker, S Jardine, Berry, McPherson, Colquhoun, I Jardine, Clark (Foster 78), Mackay, Robetson.
Dundee United: Thomson, Holt, Malpas, McInally, Hegarty, Narey, Ferguson, Bowman (Clark 72), Gallacher, Irvine (Sturrock 45), Redford.
Referee: Mr A M Roy.