The 4-0 win over Moldova put Wales top of Group D after the opening qualifiers.
With Austria, Serbia and the Republic of Ireland also in their group, Hartson urged Wales to stay focussed
"I think it's a very open group. Serbia away is a very difficult place to go, two years ago we went there and lost 6-1," said ex-Wales frontman Hartson.
"The Republic of Ireland are a very decent team now under Martin O'Neill too. Martin's teams are always very difficult to beat.
"They showed great character last night to come back for a 2-2 draw in Serbia which was result which suited Wales, of course, because you want teams to take points off each other.
"But Austria beat Georgia 2-1 and Austria have got some really good players themselves.  Next month we go to Vienna, then a few days later play Georgia at home, there is still an awful lot of work to do.
"It's such an open group, but Wales have had a great start."
Wales head to Vienna on October 6 and then entertain Georgia at Cardiff City Stadium three days later.
The national side, who are looking to qualify for a World Cup for the first time since 1958, has in the past struggled against some of the so-called smaller nations.
And despite his warnings about complacency following Wales' epic surge to the semi-finals of Euro 2016 back in July, Hartson, who earned 51 caps in his international career, believes that, under Coleman, Wales are a very different side from the one prone to fall short in the past.
"We are a confident team now and we should be with the players we have," Hartson told BBC Radio Wales.
"But there is that warning of complacency and Chris will be drumming into the lads that that won't be accepted and the players themselves will know that over the years in games like Georgia away and Moldova away we have struggled.
"But there is a different mentality in this group."
John Hartson has warned Wales to avoid complacency after their winning start to the the 2018 World Cup campaign.