Trailing 2-1 from the first leg, Uganda knew that a solitary goal would be enough at the Mandela National Stadium if they kept a clean sheet, after they scored a crucial goal in the first clash.
Massa duly delivered in the 13th minute with a superb header and it could have been more comfortable for Uganda, who wasted a host of chances to cement their advantage.
Meanwhile, a late penalty from Umaru Bangura saw Sierra Leone dispatch of Swaziland 1-0.
With the scores locked at 1-1 from the first leg, there was late drama when Sierra Leone were awarded a spot-kick with 10 minutes left, and Bangura kept his composure from 12 yards to seal a 2-1 aggregate triumph.
Rwanda were also round one victors.
Stephen Constantine's men were comfortable 3-0 winners against Libya in Kigali on Saturday, Daddy Birori the hero for the hosts with a hat-trick.
Having drawn 0-0 in the first encounter, the return leg proved a much more uneven affair, with Birori opening the scoring in the 39th minute.
He added two more goals in nine second-half minutes, as Rwanda safely progressed.
Cicero Semedo scored twice as Guinea-Bissau triumphed 3-1 over Central African Republic.
Guinea-Bissau had failed to win in their last seven matches, but Semedo stepped up when it mattered to help his team through to the next round as they won by the same scoreline on aggregate.
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.