Emsworth were on the brink of defeat following a fine last-wicket stand by Joel Eastwood (32 not out) and last man David Mitchard (11).
Coming together at 122-9 with 165 victory for victory, the tailenders took the score to 164 - before Sharma (4-30) had Mitchard stumped by Stu Parsons going for the winning hit off the last ball of their innings.
Opening bowler Rob Norris (3-12 off nine overs) had earlier reduced Sparsholt to 42-4.
Emsworth had been indebted to their own decent last-wicket stand in setting what turned out to be a winning target.
No 6 Anthony Norris (44) and last man Mike Offord (7 not out) added 36 (eventually) crucial runs.
Lewis Roberts (34) and opening bat Sharma (32) were the only other two Emsworth players apart from Norris to hit double figures.
There was also a thrillingly close finish at Lymington where the home 2nd XI defeated Hayling Island by just two runs.
Hayling, needing 221 for victory, were on the brink of victory at 215-7 with opener Mike Hallett still at the crease unbeaten on 105.
But Hallett was dismissed by Ed Freeman, with Freddie Steele run out three runs later. With the score still on 218, Hayling needed three runs off the final delivery bowled by Freeman (4-30) - but skipper Rob Cordell was bowled.
Hallett and Greg Chaplin (57) had put on 112 for Hayling’s third wicket, with the former hitting his second century in three innings - having started 2022 with an unbeaten 106 in a friendly against Southampton University.
Hallett has now scored 17 centuries for Hayling since his debut ton in 2010.