Alberto Zapater put the Rossoblu ahead after just four minutes with an unstoppable free-kick from 20 yards before Giuseppe Sculli added a second five minutes before the break.
That was enough to earn Gian Piero Gasperini's Serie A leaders all three points in their opening Group B match.
Genoa's return to major European football after a 17-year absence could not have got off to a better start with Zapater giving them the lead after only four minutes.
The Spaniard had curled in a tremendous free-kick from the edge of the penalty area in Genoa's Serie A victory over Napoli on Sunday and Slavia Prague's goalkeeper Martin Vaniak probably expected to be made to work hard when he saw him lining up shot.
However, he was still unable to prevent the goal as he could only get a weak hand to the shot, which was whipped over the wall from 20 yards.
Genoa's lead could have been short lived had Zdenek Senkerik shown more composure when put through by Adam Hlousek, but he lifted his shot over Marco Amelia and also over the crossbar from 12 yards out.
The home side continued to attack with Sculli causing plenty of problems cutting in from the left.
Vaniak parried one of his shots wide in the 27th minute before catching another powerful effort from Zapater at the second attempt four minutes later.
Genoa's persistence was rewarded in the 40th minute when Nenad Tomovic's cross from the right landed at the feet of Sculli just five yards out and he had a simple tap-in to give the hosts a two-goal advantage at the interval.
The game was practically over as a contest as Slavia's heads dropped and Genoa appeared happy to settle for a 2-0 win.
They did come close to a third, though, with Zapater diverting a Rodrigo Palacio corner onto the top of the crossbar in the 70th minute, but not even a late rally from the Czech side was able to alter the scoreline as Genoa held on to make a winning debut in the Europa League.