A first-half penalty from Juanma Delgado was enough for Hearts to end a five-game run without a Premiership win at Tannadice.
The Tynecastle side, without having to play anywhere near their best against a meek Dundee United team, reclaimed third place from Ross County with a hardfought but deserved win.
Hearts looked the more dangerous side from the outset and they opened the scoring after 15 minutes when Juanma tucked away his penalty after Sam Nicholson was tripped by Blair Spittal while bursting into the box off the left flank.
The visitors threatened again in the 29th minute when Danny Swanson had a powerful angled shot blocked by United keeper Luis Zwick after Sow had dispossessed Mark Durnan wide on the right and cut into the penalty area.
Swanson went even closer two minutes later when his sweetly-struck free-kick from 25 yards out was brilliantly tipped over by the German stopper.
United, under the charge of former Hibs manager Mixu Paatelainen for the first time, carried little threat throughout the match although they did go close two minutes before the break when John Rankin had an effort deflected behind after Neil Alexander failed to catch a John Souttar cross.
The visitors had a decent chance to increase their lead in 53 minutes when Sow powered his way up the right flank and cut the ball back for Juanma but the Spaniard’s shot from 12 yards out was blocked by former Jambo Ryan McGowan.
Hearts lost both Miguel Pallardo and Juwon Oshaniwa to injury in the 71st minute, but their plight was eased in stoppage time when Spittal was sent off for picking up two bookings in quick succession after fouling Morgaro Gomis and then Nicholson.
Dundee United: Zwick; J Souttar, McGowan, Durnan, Dillon; Spittal, Fraser (Connolly 77), Kuhl, Rankin; McKay, Bodul (Taggart 46). Subs: Szromnik, Murray, Telfer, Donaldson, H Souttar.
Hearts: Alexander, McGhee, Augustyn, Rossi, Oshaniwa (Smith 71), Buaben, Pallardo (Gomis 71); Sow, Swanson (Walker 54), Nicholson; Juanma. Subs: Hamilton, King, Djoum, Zanatta.
Referee: K Clancy