It came as a heritage watchdog warned the city faces a “massive explosion” in the population of vermin such as rats and mice as a result of rubbish piling up in the streets and spilling out of overflowing bins.
Union officials will meet with Cosla, the council umbrella body, on Tuesday to discuss an improved 5 per cent pay offer.
Cleansing workers in Edinburgh began a 12-day strike last week after turning down an earlier pay offer.
Crippling strikes are set to spread to more than a dozen other council areas, including Glasgow, Dundee and Aberdeen, from Wednesday.
Ms Sturgeon said the action comes while Edinburgh is the “centre of the cultural world” during the busy festival period.
She said: “Nobody wants to see the kind of disruption and impact of strikes that many people are witnessing in Edinburgh right now.”
Speaking during a visit to Aberdeen, the First Minister said: “We live in a really challenging time with inflation in double figures right now, which is why the Scottish Government is determined to deliver – as far as we can and to facilitate as far as we can – fair pay deals.
“We’ve provided – and the councils are the employers of the workers that you’re referring to right now – we’ve provided more resources to local authorities to try to facilitate a fairer pay deal and I’m glad to see that Cosla has now put on the table a 5 per cent pay offer and I hope that now paves the way to these issues being resolved.”
Scottish Conservative MSP Miles Briggs accused Edinburgh’s Labour council administration of an “astounding” lack of contingency planning.
He said: "More could have been done to prepare the city, such as working with private companies or providing additional bins.
"The SNP Government must now intervene to prevent international embarrassment for Edinburgh and Scotland.
"The rubbish piling up on our streets risks damaging our city's reputation. These annual festivals are supposed to be a source of pride, not humiliation.
"The SNP Government must get around the table and fix this before it's too late. They cannot stand by and watch while a situation that they created by giving councils a poor funding settlement spirals out of control."
Scottish Liberal Democrat leader Alex Cole-Hamilton said: “The effects of the strike are depressingly visible. We are seeing mountains of filth piling up.
"The blame for these strikes lies entirely in the hands of a SNP/Green Government which has cut £1 billion from local government budgets in recent years and hardly lifted a finger to tackle the cost-of-living crisis.
“We all know that August is perhaps the most important month in Edinburgh's calendar. This situation risks jeopardising the capital's reputation among festival visitors from across the UK and the world.
"Refuse workers are being hit hard by the cost-of-living crisis. They need emergency changes to the budget which will properly fund local government and support those in need.
"The Scottish Government must fund local authorities so that they can afford to give workers a proper pay rise and put an end to this sorry mess.”
Terry Levinthal, director of the Cockburn Association, the city’s heritage watchdog, said there had already been an increased problem with vermin.
He said the amount of rubbish in the Old Town and other festival hotspots had increased “exponentially”.
Mr Levinthal told the BBC’s Good Morning Scotland programme that Edinburgh had existing problems with litter and waste collection.
Asked if he was worried the situation would become "a massive public health problem", he said: "Yeah, it is – already the vermin problem has increased.”
He said the seagulls had been out “in full force”, adding: “Even before the strike there were problems.
"We had our members and stakeholders complaining, particularly in the Old Town, where we’ve seen for hospitality businesses a number of structures erected – these provide fantastic places for the city’s mice and rat population to hide away and eat all of the offerings that have been left for them, if I can put it that way.
"And we'll see, as a result of it, that in a few weeks' time there will be a massive expansion in the population of vermin because there is just so much food on offer."
Mr Levinthal said everybody had a part to play to try to manage the situation.
He urged residents not to put waste such as cardboard and polystyrene out on the street.
Mr Levinthal added: “But we also need to communicate with the tourist and event sector that there is a strike.
"And if those businesses that have commercial collection facilities can actually maybe step up, because it’s in their interests as well to present the city in the best way possible, that can actually help.
"Even when the strike ends, it’s going to take a long time for everything to be collected and got back, so the more we can do now, the better it will be in the short term.”
Alison Maclean, an industrial officer with the trade union Unite, said its local government committee “has reaffirmed that the strike action ongoing in Edinburgh and scheduled to take place in a further 14 councils continues as planned”.
She said: "There remains insignificant detail on the 5 per cent pay offer, and what this in reality means for the lowest paid workers.
"At this moment the offer from Cosla remains a vague aspirational pledge, but Unite can’t take anything to our wider membership unless we have specifics and guarantees.
"A meeting is scheduled for [Tuesday] with Cosla and we will then reassess the situation.
"Our members have taken the brave stance of taking strike action to get the pay rise they deserve and we are determined to ensure that this happens.”
GMB Scotland’s senior organiser Keir Greenaway said: “Our members want clarity from Cosla about whether this proposal comes with the assurance of a flat rate award, a key demand of the union pay claim.
“The prospect of the highest paid getting the biggest cash increases in any offer would be unacceptable, let alone one that is still well below inflation.
“Unions will now meet with Cosla officials [on Tuesday] morning, but in the meantime all existing and planned strike action remains on.”