Hibs’ top scorer netted with the first chance of the match after four minutes and Hearts, despite having the better of the second half, were unable to force an equaliser. Hearts defender Blazej Augustyn and Cummings were both dismissed within the last quarter of an hour for second bookable offences.
The hosts made one change to the side that started the goalless draw with Livingston as Kevin Thomson replaced Chris Dagnall. There was no place for injured midfielder Dylan McGeouch.
Hearts made three changes from the side that won 3-0 at Ross County last week as Igor Rossi, Augustyn and Abiola Dauda came in for Juwon Oshaniwa, John Souttar and Sam Nicholson. Midfielder Prince Buaben remained on the sidelines with a hamstring problem.
The teams emerged to a crackling atmosphere at a sold-out Easter Road and Hibs required only four minutes to have their support in raptures when Cummings nipped in to force home a David Gray cross from four yards out.
Buoyed by having netted in his fourth consecutive Edinburgh derby, the 20-year-old almost scored again in the tenth minute when he was put clean through on goal by Liam Henderson, but Hearts goalkeeper Neil Alexander raced out to make a vital save at his feet.
Hearts thought they had equalised in the 20th minute when Dauda nodded in a Jordan McGhee cross at the far post but, after a couple of seconds of celebration, it became apparent the linesman had flagged for offside.
Hibs were coping reasonably comfortably with their Premiership opponents and they threatened again in the 26th minute when Anthony Stokes, who was watched from the stand by Republic of Ireland assistant manager Roy Keane, had a low shot from just inside the box saved by Alexander after good work from Cummings.
Callum Paterson, who retained the captain’s armband in the absence of the suspended Alim Ozturk, tried to spark his team into life with a low 25-yard strike which trundled just wide of Mark Oxley’s left-hand post in the 35th minute.
Hearts came out in more purposeful fashion at the start of the second half and Rossi went close to an equaliser in the 53rd minute when he met a Paterson cross at the back post and volleyed into the side-netting.
Hearts were fortunate not to be reduced to ten men two minutes later when Miguel Pallardo, who had been booked in the first half, escaped a second yellow for a foul on McGinn. The Spaniard was swiftly replaced by Sam Nicholson.
Hearts were looking the likelier side after the break and Juanma just failed to connect with a driven Paterson cross in the 65th minute. Two minutes later, the Spanish striker had a close-range effort ruled out for offside after he bundled in from a Nicholson cross.
Hearts suffered a major setback in the 76th minute when Augustyn was sent off after picking up a second yellow card for appearing to throw the ball away.
However, the numbers were swiftly evened up when Cummings saw red two minutes later after appearing to be given a second caution for dissent.
Hearts pressed in the closing stages but were unable to force an equaliser as Hibs held firm, sparking jubilant celebrations among the home support at the end.
Hibs (4-1-3-2): Oxley; Gray, McGregor, Fontaine, Stevenson; Bartley; Henderson (Boyle 87), Thomson (Carmichael 69), McGinn; Cummings, Stokes (Keatings 90). Subs not used: Virtanen, Fontaine, Gunnarson, Dagnall.
Hearts (4-4-2): Alexander; Paterson, Augustyn, Rossi (Souttar 76), McGhee; Cowie, Djoum, Pallardo (Nicholson 59), Walker (Zanatta 84); Juanma, Dauda. Subs not used: Hamilton, Oshaniwa, Reilly, Beith.
Referee: John Beaton.