A double for Robbie Muirhead was the difference between the sides as the attacker netted his first goals for the club in near identical fashion, sliding in to finish a cross at the back post.
The game was not without its controversy, however, as Rangers saw a Joe Dodoo goal disallowed for offside with the score at 1-0.
Hearts started the same side that defeated Motherwell 3-0 on Saturday and the hosts began the brighter of the two sides, though chances were few and far between in the opening 35 minutes.
Hearts improved as the half went on and went close twice through Bjorn Johnsen. The striker brought two saves out of Wes Foderingham with his head. The latter stop, in particular, was of real top quality, as the keeper produced a fingertip effort to divert a header from eight yards over the bar.
They would find the breakthrough their play deserved a minute before the half. Johnsen, receiving a return pass from Arnaud Djoum, put in a terrific pass to the back post that found the advancing Muirhead to score.
Though Hearts started the brighter in the second period, they looked to have conceded an equaliser seven minutes in when Dodoo finished with a half-volley inside the keeper’s left-hand post. However, Harry Forrester was adjudged to have interfered with play by standing in goalkeeper Jack Hamilton’s sight-line.
Eight minutes later, Hearts were in the driving seat. Counter-attacking through Callum Paterson, the right-back slipped it through for Jamie Walker and the midfielder sent a low cross to the back post for Muirhead to put his side 2-0 up.
Rangers improved as the game entered its closing stages as they finally managed to put Hearts under some pressure, including draw a terrific goal-saving clearance on the line by Faycal Rherras, but they never looked like getting anything from the match.
Hearts move up into second place thanks to the win. And Neilson, if he goes, has managed to bookend his Hearts managerial career with league victories against Rangers. The 36-year-old was victorious in his first league match away to Rangers in August 2014.