There was nothing to separate the sides at the interval after Katie Wilkinson struck instantly to cancel out Georgia Timms’ header.
However, the Edinburgh side struck twice in eight minutes after the interval through Bayley Hutchison, rendering Laura Berry’s strike later in the half too little, too late in the context of the game.
Rangers will now shift their focus to Wednesday evening as they make the trip back along the M8 motorway to face Hibernian at Meadowbank.
Leanne Crichton named three changes to her side for the trip to Edinburgh, with a new look central defensive partnership from last month’s win over Montrose as Jess Pegram partnered Laura Rafferty.
It was the worst possible start for the Gers who fell behind from one of Hearts’ first forays into the final third as a Carly Girasoli delivery was pinpoint and was met in the penalty area by Timms who stooped to nod past Jenna Fife.
However, Rangers offered an immediate riposte as from the very next attack Mia McAulay left the Hearts full-back in their slipstream on the left-hand side before firing a dangerous delivery across the face of the penalty area where Wilkinson planted a terrific sidefoot volley back across goal and into the bottom corner.
The conditions in Edinburgh, played amid a swirling wind and spells of rain in the first-half, made the contest a midfield battle as the only other opportunity of note in the opening exchanges belonged to the home side when Timms cut inside and curled just over the top.
There were cries for a spot-kick on the half-hour mark when McAulay combined with May Cruft and pulled the trigger inside a congested penalty area, but the referee wasn’t interested in any Rangers appeals.
The away side were certainly in control of proceedings, yet Hearts’ threat on the break was underlined when Hutchison was fed after making a smart run through the heart of the Gers rearguard but she contrived to fire wildly off target.
McAulay then shimmied with intent on the left-hand side, digging out space for a cross which was too close to the Hearts goalkeeper.
The second half began slowly before slowly bursting into life when a free-kick delivery from the left was headed on by Pegram on the edge of the penalty area, but Berry couldn’t quite sort out her feet to scoop an acrobatic finish goalwards.
Indeed, the left flank appeared the most likely route to goal for Rangers as McAulay once again showed her value by working the space for a cross which almost sailed over the goalkeeper and into the back of the net.
But for all of the visitors’ endeavour in the contest they fell behind on the hour-mark when Fife did tremendously to deny Olufolasade Adamolekun’s initial effort from a pull-back but could only watch Hutchison slam home the rebound from point-blank range.
Fife was then forced into an outstanding stop on the full-stretch when Timms produced a sumptuous overhead kick from the edge of the penalty which looked to be dropping under the crossbar before the goalkeeper’s intervention.
Two soon became three for the Edinburgh outfit when a corner was floated dangerously into the penalty area where Hutchison, on the scoresheet moments earlier, rose highest to glance a pinpoint header into the far corner.
Crichton turned to her bench by introducing Quinty Sabajo in place of McLeary, with the game being played almost entirely in the Hearts half as the Gers sought a response.
Hope was resored and the arrears reduced with ten minutes remaining when another wicked delivery from McAulay reached Camille Lafaix at the far post and, after the American couldn’t bring the ball into her vicinity, it sat nicely for Berry to poke beyond the Hearts goalkeeper.
The intensity suddenly rose as Lafaix went close herself from a cross moments later.
But Hearts stood tall in the face of the away side’s pressure in the dying embers as time eventually ran out on the Gers in their search for an equaliser.
RANGERS: Fife, Austin, Pegram (Hay, 84), Rafferty (Eddie, 60), Docherty, McLeary (Sabajo, 75), Cruft, Wilkinson, Brookshire (Lafaix, 60), McAulay, Berry
Subs not used: Kelly, Connolly-Jackson, Black
HEARTS: Vilao Dias, Ypema, J. Husband, E. Husband, Fulton (Andrews, 55), Timms, Adamolekun (Slattery, 80), Wade (Hunter, 80), Dolan, Hutchison (Johns, 70), Girasoli (Forsyth, 80)
Subs not used: Johnstone, Jardine, Richards, Chomczuk
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