Lillestrom's Nosa Igiebor plucked a brace, Ekigho Ehiosu scored on a second debut and Haruna Lukman could afford to miss a second half penalty after scoring in the first half.
Faced with five full internationals on the Ethiopia side, including their national team first choice goalkeeper, Eguavoen started with a side chock-full of Europe-based players.
The home side needed a big win to start their campaign on the right footing, and they did not disappoint.
Odion Ighalo, much-maligned after a poor World Youth Championship two years ago in Egypt, showed he had put all that behind him, as he opened the scoring after just 20 minutes.
Lukman followed soon after, making a late arrival into the box to apply a simple finish after dinking past one defender.
Nosa Igiebor made it three soon after and wasn’t long before he crashed in his second, and Nigeria's fourth.
When the impressive Nnamdi Oduamadi won a penalty with his trickery, Lukman could have made it five, but skied his penalty.
But Ehiosu, who scored on his Super Eagles debut, showed he could be equally impressive at a lower level, and put additional gloss on a comfortable scoreline with minutes left.
The second leg comes up in two weeks time.