In the searing heat in Yerevan, the Scottish champions put themselves into a commanding position ahead of the return match in Glasgow thanks to goals from French striker Odsonne Edouard, winger Forrest and midfielder Callum McGregor.
Brendan Rodgers’ side need to negotiate four qualifying rounds this year to reach the lucrative group stages for the third year running.
At Celtic Park next Wednesday night, the Hoops will, barring a remarkable turnaround, book their place in the second qualifier against Valur Reykjavik or Rosenborg.
Forrest acknowledged the importance of Celtic’s two late goals - he scored in the 81st minute before setting up McGregor for his 90th minute strike - and believes they can build on that performance.
“We finished the game strong,” he told BBC Scotland. “We got another two goals and I think we could have scored another couple as well so I’m glad we kept the clean sheet and we will take that into next week. I think we will be stronger as well.
“We knew how hot it was going to be. We couldn’t play the way we normally do but I think that was good for our confidence, to show we can go out in different conditions and play good football.”