After Katie Boulter's victory over beat Valentini Grammatikopoulou, Konta knew that victory would give Great Britain their second win in a row following the 3-0 win over Slovenia.
However, it was far from plain sailing as an impressive Sakkari took the opening set, breaking Konta twice in the process.
And things looked as if they may go from bad to worse when Konta was broken in an epic game at the start of the second stanza.
Konta held firm however and she then broke Sakkari twice and held in another marathon game in the sixth.
Both players had issues with the officials and in the third set there were a couple of tense and closely fought games but in the end Konta triumphed for a crucial win.
The victory for Great Britain now sets up a clash with Hungary to see who will secure top spot in Pool A.
"I don't think you have to ask! [How hard the match] It was a great match and consistently on her behalf." Konta said afterwards.
"I had to work very hard to bring my level up because she raised her level and was playing far better than me. I had to really dig deep and make it a battle.
I'm just so happy I was able to use your support once again. It's hard to explain what it feels like to be in front of all your support.
"It's quite special. I don't think I will experience anything like it again in my life."
Later in the evening, Harriet Dart and Katie Swan ensured it was a whitewash for the Brits against Greece, winning their doubles rubber in straight sets.
The pair were too strong for Anna Arkadianou and Despina Papamichail, winning 6-1 6-4.
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Katie Boulter battled through to beat Valentini Grammatikopoulou 6-3, 4-6, 6-3 to give Great Britain a 1-0 lead over Greece in the Fed Cup on Thursday.
On Wednesday Boulter gave Great Britain the perfect start in their 3-0 win over Slovenia as she hammered Kaja Juvan in straight sets.
She opened up against Grammatikopoulou in perfect fashion as she broke her opponent in the opening game of the match.
Having taken the opening set Boulter was then broken in the third game of the second set and was broken again as Grammatikopoulou levelled up.
Katie Boulter of Great Britain reacts during Day Two of the Fed Cup Europe and Africa Zone One Group I at University of Bath on February 07, 2019 in Bath, England.
Image credit: Getty Images
Grammatikopoulou then broke Boulter in the first game of the decider, leading to nervous moments for the crowd at the University of Bath.
However, the turning point came during two titanic games as Boulter first broke back to level at 2-2 and then served out to take the lead.
Another break came later as Boulter sealed what could be a vital victory.