A goal in each half from Jamie Walker took the Edinburgh club back to the summit after Dunfermline’s draw at Raith Rovers.
In an efficient overall display from Robbie Neilson’s side, Andy Irving’s passing in midfield stood out and was a central factor in Walker’s second.
Morton had conceded just two home league goals prior to this encounter but they had young goalkeeper Aidan McAdams to thank for not losing by a bigger margin.
Hearts created a number of other opportunities, although in the end they were content leaving Greenock with a 2-0 scoreline in their favour.
Despite the extensive time since these clubs last met, Cappielow has changed little over the years. It is a throwback to a largely bygone era when Scottish football was full of wooden stands, concrete terracing and corrugated iron roofs. And, of course, the giant crane way behind the west end of the ground sets it apart visually.
Hearts needed to overcome the humble surroundings and dig out a result after a shock Betfred Cup loss at Alloa last weekend. With Northern Ireland striker Liam Boyce on the substitutes’ bench, Craig Wighton got a chance at centre-forward.
Morton’s 5-4-1 system is set in stone under manager David Hopkin and their rigid defensive shape was evident immediately. However, they also came closest to scoring in the early stages through Sean McGinty.
Walker stung the palms of the Morton goalkeeper Aidan McAdams on 22 minutes. Several other Hearts attempts landed off target or were blocked by flying bodies in blue-and-white hoops.
McAdams produced an excellent 35th-minute parry from Wighton’s volley, which was struck instinctively as Michael Smith’s loping forward ball dropped over the striker's shoulder. Moments later the visitors’ concerted pressure paid dividends.
Lee’s ball found Walker on the left side of the penalty area for a right-footed volley into the ground, and the ball spun up as it soared into the far corner of the net.
Hearts’ advantage would have doubled before the interval but for another important intervention from McAdams. He touched Olly Lee’s shot onto the crossbar at full stretch.
Minutes after the restart the second did arrive, thanks in no small part to Irving. His pinpoint crossfield pass landed perfectly for Walker to chest down, evade an opponent, cut inside another and drive a low finish across McAdams into the net.
Irving played in a the middle of Hearts’ three midfielders for most of the afternoon and was able to dictate periods of play with his cultured left foot. The second goal put his team in command of what was a potentially-awkward fixture.
Morton showed more attacking intent for the rest of the game but their industry was in vain as Hearts held on for three points and a welcome clean sheet.
Morton (5-4-1): McAdams; Omar, Fjortoft, McLean (Salkeld 60), McGinty, Strapp; McPake, Jacobs, McAlister, Muirhead (Nesbitt 77); Maciver (Oliver 75).
Unused subs: Millar, Orsi, Lyon, Colville.
Hearts (4-3-3): Gordon; M Smith, Halkett, Berra, Kingsley; Lee, Halliday, Irving (Haring 75); Naismith, Wighton (Roberts 77), Walker.
Unused subs: Stewart, White, Boyce, Popescu, Henderson.