Heart of Midlothian 4 Dundee United 0
For a while they looked in danger of stumbling to another shock home defeat in their first appearance at Tynecastle since St Mirren's famous win three weeks ago.
Dundee United created but failed to take several chances before ill fortune plus ruthless Hearts finishing settled the outcome.
United's first near miss came in the ninth minute when Noel Hunt fired in a half-volley which was blocked by the elbow of keeper Craig Gordon.
Midway through the first half Stuart Duff was allowed to leap unchallenged and power in a looping header which clipped the woodwork with Gordon beaten.
Four minutes later Duff's diving header forced another reflex stop from Gordon and Hunt was controversially ruled offside as he hammered home the rebound.
Hearts then went ahead with a highty fortuitous goal. Lithuanian Andrius Velicka did well to get away a left-foot shot from a tricky angle, but keeper Derek Stillie looked certain to save it until the ball took a wicked deflection off full-back Alan Archibald and flew high into the net.
Stillie made a superb one-handed save to deny record signing Mirsad Beslija before the Bosnian winger delivered a perfect cross from the right which Finnish striker Juho Makela sidefooted in from close range.
United kept battling, but Archibald's foul on Beslija allowed Paul Hartley to make it 3-0 from the penalty spot and substitutes Roman Bednar and Jamie Mole then combined to leave the latter with a simple tap-in.