UCF trailed 21-10 before a late score, but following a Tigers punt, Knights receiver Jeff Godfrey's fumble with 2:26 left allowed Missouri (3-2) to hang on.
UCF (2-2) controlled most of the game, using five sacks and a huge advantage in time of possession to take an early lead.
But miscues on defense and special teams unraveled it all, and the Knights sputtered on offense down the stretch in their bid to defeat their first Southeastern Conference opponent at home.
Coming off a dismal passing performance last week at South Carolina, Franklin was efficient, going 19 for 30 for 257 yards.
Kendial Lawrence added 104 yards rushing and a touchdown for the Tigers.
Knights quarterback Blake Bortles kept his team in the game, connecting on 29 of his 43 passes for 267 yards and two touchdowns.
Prior to Murphy's punt return, the Knights looked to be taking control of the game, leading 10-7 and coming off an interception and fourth sack of Franklin early in the third quarter.
But the slide continued with Missouri capitalizing on a short punt and needing just three plays to take its 21-10 advantage with 9:31 to play in the game.
Offense was plentiful in the first half, with both teams using big plays through the air to reach the end zone.
Missouri had several promising drives stall early before getting on the board via an 80-yard touchdown strike from Franklin to Dorial Green-Beckham. That gave the Tigers a 7-3 lead in the opening minutes of the second quarter.
UCF defensive back A.J. Bouye briefly had Green-Beckham locked up by the legs near the 50 before slipping to the turf. Green-Beckham then sprinted untouched into the end zone.
The play, the Tigers' longest from scrimmage this season, also was a career-best for Franklin.
The Knights responded on the ensuing drive, with Bortles using a 41-yard pass to Rob Calabrese to set up an 11-yard touchdown pass to Quincy McDuffie.
Following a series of punts, UCF got hot again and drove the ball from its own 12 all the way inside the Tigers' 10 with less than 2 minutes to play in the half.
A pass interference call on Missouri gave the Knights first-and-goal on the 3. But after a stuffed run and incomplete pass, the opportunity was squandered as Bortles was sacked, and time expired before the Knights could run out their field goal team.
Despite the final miscue, the Knights' offense had one of their best opening halves of the season with Bortles completing 12 straight passes at one point and 18 for 22 overall for 156 yards. Johnson also averaged nearly 8 yards per carry in racking up 84 yards rushing.
The Tigers were successful at times with their hurry-up offense, but UCF's defense also helped keep Franklin off balance, sacking him three times in the opening 30 minutes.