Nicknamed 'Chicharito' (Little Pea), the diminutive Mexico striker, who joined from Manchester United last month to replace Son Heung-min who left for Tottenham Hotspur, scored the Germans' third goal, sliding his low shot under keeper Sergei Chernik.
Admir Mehmedi and Calhanoglu had earlier scored for Leverkusen either side of Nemanja Milunovic' equaliser before Calhanoglu scored from the penalty spot in the 76th minute.
Leverkusen are top of Group E on three points with Barcelona and AS Roma drawing 1-1 in the other group match.
The Germans, who have had a rocky start to the domestic season and have lost their last two Bundesliga matches, needed only four minutes to take the lead with Mehmedi curling a shot from inside the box after Borisov defenders failed twice to clear the ball.
Their joy lasted only nine minutes despite coming close once more when Hernandez failed to connect with a low cross in front of goal.
Serbian Milunovic silenced the home crowd with a goal against the run of play after being given too much space to see his deflected header sail into the back of the net.
The hosts continued to press and Mehmedi came agonisingly close but his shot scraped the post as it went wide. They bounced straight back after the break, however, with Calhanoglu's shot taking a deflection to fly over the keeper and into goal in the 47th minute.
Hernandez then struck 12 minutes later after a fine pass from Kevin Kampl to tip the game firmly in the Germans' favour before Calhanoglu sealed a deserved victory with a well taken spot kick.
(Reporting by Karolos Grohmann, editing by Pritha Sarkar)
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