The wounded security official was identified as Hassan Jarbouh, the deputy chief of the Rafah branch of the Preventive Security Service.
There was no immediate claim of responsibility for the drive-by shooting but security forces said they were searching the area for the attackers.
"There should be an end to all these incidents ... . We do know who did it, but I will not name them at the moment," Youssef Abu Siyam, a senior security officer, said.
The confrontations between the two sides intensified nearly two weeks ago when the three young children of an intelligence officer were shot dead, and intensified after Abbas' announcement last week that he is seeking new elections.
Hamas has accused Abbas of trying to topple its 10-month-old government.
In an interview aired by Aljazeera on Saturday, Khaled Mishaal, Hamas politburo chief, urged all Palestinian factions, including Fatah and Hamas, for a comprehensive dialogue leading to the formation of a unity national government.
"Hamas is ready to go to this dialogue unconditionally," he said.
Commenting on Abbas's call for early legislative and presidential elections, Mishaal said: "Elections should not be held without a national consensus."
He warned of yielding to foreign pressures, saying "response to these pressures was behind the flare-up of inter-Palestinian disputes".