Uganda's lone try came in the 70th minute.
The Monks have an arduous quarter-final task against Kabeberi Sevens champion KCB Rugby.
Familiar foes clash at Stade Maurice David in France for place in the semis.
Will Dala Sevens produce a new winner?
Uganda Cranes Rugby back Philip Wokorach’s trademark sidesteps are well known locally.
Captain Dan Sikuta said they have a good opportunity of qualifying.
The Simbas are due to leave the country Wednesday morning.
Kenya will meet Uganda in the quarter-finals on July 2 at Stade Maurice David, Aixen.
The government is required by law to fund all national teams in international assignments.
The provisional squad of 34 will be reduced to a final 28.