A 3-0 win at Kilkenny's Buckley Park ground on Thursday night sealed progression, after a brace from Keogh in Baku last week had given the Irish a 2-1 advantage.
Cork City's Roy O'Donovan and Lutons Kevin Foley were on target in the second leg, as well as Keogh.
20-year-old Keogh's goal, which made it 3-0, came after he made the most of a mistake from defender Elnur Abbasov to slot home from close range.
The Dublin-born striker had also created the opening goal, crossing from the right for O'Donovan to head home.
The Republic will now place Belgium and Greece in a group stage in September, with boss Don Givens in confident mood.
He said: "It was a good solid performance. The lads were very professional and passed the ball around well. I thought we'd score more goals towards the end, but we did the job.
"It was just what we asked of them. We knew the job had to be done and there was some very good stuff from us.
"Belgium and Greece will certainly be a step up in quality so we have a lot to do but hopefully we'll have some players back who were missing for this one.
"We've got the Belgians at home first on September 3 and if we could win that one it would give us a real chance in the group. We are due to play the Greeks two days later.
"The confidence is good and the lads are hungry to perform. If we can keep at this level by creating chances up front and keeping it tight at the back I think qualification will come by itself.
"We're not obsessed by qualification but we're going to give it our best shot."