Johanna Konta faces Maria Sakkari next on Thursday, before a doubles rubber.
A Konta win would clinch the tie for GB, who beat Slovenia 3-0 on Wednesday.
Boulter said "That was a little bit dramatic," ."I put absolutely everything on the line there. I am a fighter."
An initially-calm Boulter looked assured in the first set, breaking Grammatikopoulou's serve twice, but the Greek 21-year-old fought back, showing more aggression to clinch a tight second set.
Boulter looked unsettled after the umpire controversially overruled a line judge's call when the Greek player was serving for the second set. The home fans thought the ball had landed out, which would have handed Boulter two break-back points.
However, backed by a lively sold-out crowd of 1,800 at the University of Bath, as GB play on home soil this week or the first time in 26 years, she recovered her form as the match entered a third hour.
Boulter told fans: "If it wasn't for you guys, I don't know if the result would have been the same,"
Ultimately, a gutsy victory continued Boulter's 100% record in Fed Cup tennis and kept Anne Keothavong's side on course to top Group A.
How can Great Britain reach the World Group?
Having been outside the top two tiers since 1993, GB are bidding to reach the World Group II play-offs for a third consecutive campaign.
There are eight teams - split into two groups - taking part in the Europe/Africa event in Somerset until Saturday, when the group winners will meet to decide who progresses to April's play-off event to fight for promotion to the second tier.
From Group B, either Croatia or Serbia - who will meet on Friday - will be involved in Saturday's decider, after they both won again on Thursday to make it two wins from two.
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