The Colombian striker netted with just eight minutes to go, after Juan Vargas had cancelled out Antonio Di Natale's openerUdinese a 2-1 victory over Fiorentina in their Coppa Italia semi-final, first-leg clash at the Stadio Friuli on Tuesday night.
In slippery conditions, a late strike from the Colombian substitute strike sunk the visitors.
Captain Antonio Di Natale was the man to hand Udinese the lead in the first leg, scoring from close range nine minutes before the break to register his first goal of this year's competition.
However, when the 36-year-old was dispossessed on the stroke of half-time, Fiorentina quickly got the ball to Juan Vargas, who delivered a thumping effort in off the crossbar to equalise in style.
Just when the game looked to be heading for a draw, Muriel produced a rasping strike that beat Neto and put the home side on top.
Modibo Diakite, a January loan signing from Sunderland was handed a first start for Fiorentina, while 17-year-old Simone Scuffet continued in goal for Udinese after making his debut in Saturday's 2-0 victory over Bologna.
Despite a lively opening, neither side were able to force a shot on goal in the opening exchanges, limited instead to wayward crosses and misplaced final balls.
Gabriel Silva's deflected cross from the left after 19 minutes caused Neto an uncomfortable moment before Vargas had Fiorentina's best effort, forcing Scuffet to tip his shot round the post.
Those opportunities brought the game to life somewhat, as both sides attacked regularly without creating a host of chances.
Giampiero Pinzi was next to threaten Neto's goal, with the Brazilian parrying his effort into the path of Roberto Pereyra, who could only find the side-netting with his follow-up effort.
However, veteran striker Di Natale put Udinese ahead, tapping home Pereyra's cross after Silvan Widmer had helped drive a quick break from the hosts.
Di Natale is set to retire at the end of the current season and he looked to have ensured his side went in ahead at the break.
However, Vargas' fierce left-footed drive from outside the area levelled the scores, crashing in off the crossbar to leave Scuffet helpless and register what could prove to be a crucial away goal.
After the interval, Fiorentina began well, with Joaquin's volley well blocked by Silva from Borja Valero's cross before Alessandro Matri's improvised chest and volley threatened Scuffet's goal.
Muriel eventually got the winner eight minutes from time, when his long-range strike crept in off Neto's right-hand post.
It extends Udinese's winning run against Fiorentina to three matches in all competitions and boosts their hopes of revenge for their Coppa Italia round of 16 exit against the same opponents last season.