Goals from Ibrahim Salifu and Isaac Agyenim Boateng in each half ensured a comfortable victory for the Phobians who were winless in five games across all competition coming into this 1st league playoff encounter.
Hearts started the game with intensity and Kofi Kordzi had a glorious chance to draw first blood for the home side in the second minute but skied his effort wide from close range after being teed up by Isaac Mensah.
Hearts were the much better side in the opening minutes and deservedly scored the opener through Salifu Ibrahim who headed in Afriyie Barnieh's low cross in the 17th minute of the game.
JS Saoura striker Mohamed Elamine Hammia had an opportunity to strike back for the visitors three minutes later but shot wide.
The Phobians looked a very confident side in this encounter despite their recent setbacks in the Ghana Premier League and we're in control of the game.
Hearts despite their dominance were a bit lackadaisical in defence and Mohammed Alhassan's indecisiveness gifted JS Saoura striker Mohamed Elamine a chance at goal on the half hour mark but goalkeeper Richard Attah dealt with the situation comfortably.
Hearts attacker Isaac Mensah who was largely silent in the game had a chance to double the lead for the Phobians but his curled effort went wide.
Despite the delay in return by the Algerian side, the second half did start with Hearts on the front foot in search of the cushion goal.
JS Souara seemed content with the scoreline despite trailing as they started displaying some delaying antics to despair of the Hearts fans.
Daniel Afriyie came close to scoring the second goal for the Phobians but his header in the 55th minute narrowly skewed wide.
Kofi Kordzi nearly found the top corner with a well struck free kick by JS Souara goalkeeper Zakaria Said pulled off a stunning save to deny the attacker.
Hearts had a flurry of chances falling their way to increase their lead and best of the lots fell for Isaac Mensah who shockingly missed from close range with the goal at his mercy.
Substitute Isaac Agyenim Boateng deservedly scored the cushion goal for Hearts in the 71st minute when he struck home from close range after a delightful cross from William Dankyi.
JS Saoura goalkeeper Zakaria Saidi was sent off by Egyptian referee Mohamed Abel Elsaid Hussein after a second bookable offence having wrestled Afriyie Barnieh to the ground.
Hearts couldn't add onto their 2-0 lead despite getting the man advantage as they secured a slim advantage going into the second leg which is slated for December 5, 2021, in Bechar, Algeria.