Striker Kevin Doyle put the visitors ahead from close range after eight minutes and a deflected Aiden McGeady goal all but put the game beyond the world's joint-lowest ranked team after just 20 minutes.
Ireland failed to keep up their frantic early pace but with Andorra unable to trouble Shay Given's goal, Giovanni Trapattoni's side kept an eighth successive clean sheet and maintained their unbeaten group stage away record.
Russia's 1-0 win in Slovakia means the group leaders just need to beat Andorra at home on Tuesday to qualify in top spot, leaving Ireland to scrap it out for second spot with an Armenian side who have scored 11 goals in their last three games.
With neither side set to finish as the best second-placed team, which brings an automatic berth in the finals, they will likely be vying for a play-off spot with Armenia battling to reach a major championship for the first time.
Ireland won 1-0 in Armenia in the opening game of the group last year but Vardan Minasyan's side have improved dramatically since, thrashing Slovakia 4-0 away from home last month.
"Everybody was saying after our 1-0 win in Armenia on the first game that it was a horrific result but it just shows you now, it's turned a little bit," man of the match McGeady told national broadcaster RTE.
"Armenia are on a good run of form at the minute... it's going to be a tough game on Tuesday but one we want to take three points from."
Ireland left-back Stephen Ward will be suspended for the game after being booked for diving on Friday but centre-half Richard Dunne, whose man of the match display helped scrape a draw in Russia last time out, returns from a one-match ban.
