With the Republic of Ireland breathing down their necks in Group D, Wales needed to consolidate second spot in the Georgian capital and Tom Lawrence's strike shortly after the interval saw them to an important victory.
They next face the Republic on Monday in Cardiff, where the runners-up spot and a potential play-off to reach Russia next year may be on the line.
Ramsey told Sky Sports: "We're used to these situations, we put ourselves in these situations time and time again and we find a way, that's what's so special about this team.
"We wanted to finish off with two wins, we're halfway there. We wanted to go back to Cardiff with the three points and it's going to be a big game on Monday night.
"We know our fans are going to be magnificent on Monday and hopefully we can give them something to cheer about."
It was the first time that Wales had managed a victory in 11 games when their star man Gareth Bale had been absent, the Real Madrid forward missing out with a calf strain.