But while those 80 minutes in Round 18 will forever be remembered by Aironi aficionados, Phillips insists it is the work that has been done behind the scenes that really paved the way for Saturday's success.
"We've worked hard over the last six months and I think we've definitely been making progress. Last Saturday reinforced that," said Phillips, who took over from Franco Bernini as head coach in early November.
"We've just focused on improvements. We haven't put ourselves under pressure about chasing our first win.
"We haven't put all our emotions into one big, big game because that's the one we feel we could win. We've broken down what we need to do to improve in order to win and we've worked on those things.
"We value small improvements and we value people who work hard. We're talking a one-per-cent improvement here and a one-per-cent improvement there and it's only a matter of time before all those improvements add up.
"It's obviously a relief because you want to win - you play this game to win - but the preparation has been purely on improvements. We always felt that if we could improve enough, we would get that first win."
And while the league table may have told a harsh story prior to last weekend, a closer look at Aironi's season backs up Phillips' own assessment.
After what was undoubtedly a difficult start to their debut campaign, Aironi have been quietly on the rise. They may sit at the foot of the standings but their performances in recent months have been full of marked progression.
No side has picked up a try-scoring bonus point against Phillips' men in their past nine fixtures, while Aironi themselves have secured four losing bonus points and have only once been beaten by more than two scores in that three-month period.
But while the facts make it clear that Aironi have made vast improvements to the defensive side of their game, Phillips says that it has been a case of evaluating the whole package rather than channeling all their efforts into one specific area.
"To be honest, we've been looking at all areas," added Phillips, a former police officer and dual code international for his native Wales.
"We're not in a position that we can look at one specific area of weakness - we saw that we had some good structures in place in all areas and that we just need to get better at delivering them.
"A great way to improve all areas is to improve the mentality of the players. It's about improving what we do in the working week and improving the focus of the week. It's about improving players' self belief and all those key values that make up a team."
Aironi are proving increasingly difficult to break down
Becoming accustomed to the intensity of the Magners League has also been pivotal in bringing about the improvement that Phillips talks so honestly about.
With the majority of Aironi's squad coming from the old Italian Super 10 tournament where big games would be followed by far less pressurised encounters, getting to grips with the constant physicality and continually high demands of Celtic competition understandably took time.
"The mentality of this competition was something that took a while for everybody to settle down and get used to. We needed to appreciate that the challenge at the end of every week is the same.
"It's definitely something they're getting used to. I think the working work is complementing that. We're producing good, smart detail through the week and we're working hard with a specific focus towards whoever we're playing.
"The players' awareness of the challenge of this competition has been heightened. We're a lot smarter as a staff as well.
"So I think a combination of everything has certainly allowed the players to be more conducive to and more settled on the Magners League challenge."
The latest installment of that challenge comes this Saturday when Aironi host last season's Play-Off semi-finalists Glasgow Warriors at the venue of their historic success some seven days earlier.
And instead of going overboard about their first Magners League win, Phillips and co have already turned their attentions back to their primary goal of continual improvement.
"Obviously everyone was pleased with the result but our debrief from Connacht showed that there were quite a few areas that didn't meet the standards we set as a group," continued Phillips, who speaks with plenty of intelligence and no shortage of passion about his adopted side.
"We know that Glasgow, especially now that they're strengthened by the international guys, are going to be a huge challenge.
"They're a strong team, a really physical team and we know it's going to be tough game. Again, our focus has been on improving the areas we need to improve because in no way are we the finished article."
Aironi will be looking to take another leap forward this weekend
But while last week has now been forgotten as the focus switches towards the battle with Glasgow and nothing else, it is worth remembering the impact that Saturday's fine win has had on the whole of Aironi.
As an entirely new entity, the Aironi faithful have had it tough at times this term but they and the players have never let their heads drop.
Anyone who has spoken to those within the Aironi camp or to the disciples at the Stadio Zaffanella will tell you the same thing: the spirit of Aironi has always been first class.
"Our attitude has never dropped below a positive level. We haven't really given the players opportunities to drop their heads because we've focused on the positives and we've worked hard on all areas.
"To be honest, it felt like a job well done against Connacht. I'm not saying it was an anti-climax because it wasn't, but the preparation has been very good going into the games and we knew that if we prepare well and always work hard to implement game plans and improve our standards then we'd have an opportunity to be successful.
"Because we'd worked so well throughout the week and our preparation was good, to get a result just rounded off a good week's work.
"It was a great occasion for everybody, including the supporters. We've had to go through a tough period and we're starting to repay their faith in us.
"They're a knowledgeable crowd. They're the same as any other supporters in that they appreciate hard work.
"We've had some poor games and we can't hide from that fact but we competed well against Munster at home and lots of other games like Ulster at home earlier in the competition, so there's been a lot of positive reasons for the supporters to believe that this team has something to offer."
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