The game started at a frantic pace, with the hosts getting on the front foot early on.
James Collins came close for the away side a minute later, but he could only head narrowly wide.
Hibernian began to push forward in search of an equaliser, but it was Hearts who looked more dangerous as Carrick and Stevenson continued to link up well.
The visitors began to see much more of the ball and came close through a curling shot from Danny Handling that whistled just past the post.
Hearts almost doubled their lead just before half time, but Sam Nicholson's rasping shot was kept out well by the goalkeeper.
Hibs piled on the pressure in the opening minutes of the second half and created an opportunity for Alan Maybury, who volleyed over from 20 yards out.
Carrick's movement continued to trouble the Hibs back line as Hearts looked to increase their advantage.
Hibs dominated the closing stages as they pinned Hearts into their own half, but the home side's defence stood firm.
The away team went down to 10 men on 89 minutes when Maybury brought down Billy King as Hearts broke and received his second yellow card.
Hearts broke again two minutes into stoppage time and wrapped up the points through King, who calmly rounded the goalkeeper and finished to make it 2-0.