A series of unforced errors and a double fault saw Gauff drop serve early on in a one-sided first set, and she shanked a forehand wide when handed the chance to break back.
More misses on her forehand gifted Muchova a double-break lead and the Czech kept her level consistently high to serve out the first set with an ace.
Having taken a bathroom break before the second set, a re-energised Gauff returned with more aggression, and she eventually found a way past Muchova on her ninth break point with a sublime cross-court backhand winner.
The momentum stuck and Gauff flew through the next four games to force a deciding set to the delight of the crowd – the majority of whom appeared to be backing the seventh seed.
The tension rose as the third set unfolded and the pair entertained the 15,000-strong crowd on Centre Court with top-quality groundstrokes and sensational quickfire play at the net.
It was fitting, then, that the match was decided with a tie-break, and from 4-1 down, Gauff somehow found a way to move level at 6-6.
The twists and turns continued as Muchova received a warning for a time violation at 8-8 on serve – and she immediately fired long to hand Gauff the first match point.
But the crowd were left gasping in anguish as Gauff, approaching the net, swiped the ball into the tape.
To add to the drama, Muchova slipped when she had the chance to convert her first match point, and Gauff produced an audacious lob to cling on.
But the young American could not salvage the next one, planting a low forehand into the net after an extraordinary two hours and 35 minutes.
Muchova, who had clutched her ankle and gestured towards her side in pain during the final set, looked drained as she raised her arms in celebration.
A wrist injury sidelined her for 10 months last season, but she has rediscovered her finest form this fortnight to beat three former Grand Slam champions on her way to the final.