Sharif helped the Fifers recover from a shaky start at Inverleith by compiling a superb century, his second of the campaign.
Falkland had lost openers Bryn Cleaver and Connor Ryan to veteran seamer Craig Mackellar, before Sharif launched his rescue act. He took the attack to the home bowlers en route to a magnificent ton which included nine boundaries and added a further boundary and one maximum before being bowled by Mackellar for 124.
Falkland posted a challenging 218-8, but there were more heroics to come from the Scotland all-rounder as his four-wicket burst helped reduce SMRH from a threatening 119-3 to 144 all out.
Aberdeenshire stepped-up their title bid with victory over Carlton at Mannofield. Chasing a revised target of 149 after several rain interruptions, the Dons were given a good start by Chris Venske and Callum Howard before the former was run out for 32.
Howard followed for 27, but overseas duo Hayes van der Berg and Harsha Cooray guided the hosts to victory with 37no and 34no respectively.
Earlier, Carlton, having been sent in to bat by Tyler Buchan, were soon in trouble at the hands of the Shire skipper. He first found the edge of Hamish Gardiner's bat to remove the Scotland opener for six and then bowled Preston Mommsen for three.
At 21-2 in the ninth over, the visitors were in need of some solid resistance and it came from Ali Evans. The former Derbyshire fast bowler was promoted to the top of the order after surviving for two hours as a nightwatchman for Scotland A against Yorkshire 2nd XI last week and responded in style, repelling the new ball with a dogged knock that brought 32 runs from 63 deliveries.
The arrival of Cooray into the home attack brought his downfall as he was bowled by the Sri Lankan. Evans' contribution was to prove the second highest for his team, bettered only by Fraser Watts' battling 40no.
The former Saltires opener was frustrated at regular intervals by the loss of a string of partners. The departure of the promising Ruari Macpherson after a bright 19 - one of three run-outs in the innings - was a particular blow.
Cooray added another two wickets to his haul before Buchan returned to polish off the tail, ending with 4-35 as Carlton were dismissed for 158 in the 39th over in an eight-wicket loss.
Watsonians also left the Granite City empty-handed after a four-wicket defeat to Stoneywood-Dyce.
On the other side of the country West of Scotland's miserable season continued with a seven-wicket defeat to East Kilbride at Torrance House.
West's Lucas Farndale was run out in only the second over after some sharp work by home capatin Josh Johnston. Worse followed when Jatinder Singh, having survived for 19 uncomfortable deliveries, was eventually beaten by the excellent Syed Shah and the batsman's false shot found the safe hands of Peter Bocock.
Shah struck again to remove Ed Clarke and, at 14-3, the visitors were in deep trouble. The stoic Stewart Leggat dug in and reached 27 while player-coach John Blain added a hard-hitting 37.
However, West were dismissed for just 140, giving the hosts a relatively comfortable run-chase, especially when Moneeb Iqbal hit his second consecutive league half-century.