The Colombian's fourth goal of a mediocre season handed Francesco Guidolin's side a slender lead to take into next Tuesday's second leg in Florence, with Juan Vargas grabbing a precious away goal with an amazing long-range shot of his own after Antonio Di Natale had opened the scoring.
The win means Udinese are still on track for their first major trophy, which would make up for a poor league campaign that sees them struggling in 15th spot in Serie A with 23 points from 22 games, six points away from the relegation zone.
Di Natale gave the hosts the lead against the run of play nine minutes before the break, the former Italy striker tapping in his second goal in a week from Silvan Widmer's cross at the end of a sweeping counter attack.
Fiorentina were screaming for a penalty a minute later when Alessandro Matri's flick in the area appeared to touch Maurizio Domizzi on the elbow, but referee Carmine Russo waved away the visiting side's furious appeals.
Vargas shot them level with a piledriver a minute before halftime after Di Natale was dispossessed on the halfway line and they looked more likely to grab a winner in the second half.
However, Muriel thumped a beautiful finish in off the post to delight the majority of the hardy crowd who braved the torrid conditions and keep the home side's season alive.
Roma host stuttering Napoli on Wednesday in the second semi-final, with Rudi Garcia's Rome side hunting a 10th Italian Cup triumph.