The Kirkcaldy visitors appeared to be cruising to victory when they needed 56 runs to win with eight wickets remaining.
But a spirited comeback by Gala saw Dunnikier needing six runs to win off the final over with just one wicket left.
They could only manage five, leaving both teams to take 15 points each.
Batting first, Gala got off to bad start as Jack Halls was dismissed for two in the first over. Brothers Robbie and Josh Irvine led a comeback until the latter was out for 14.
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Further wickets fell to leave Gala 97 for five at the halfway point of the innings. But South African Ash Rhode fought back, top-scoring with 37 before being adjudged lbw.
Gregor Mein (20) and Robbie Pringle (14) pulled Gala up to 177 but, with the side dismissed with more than 10 overs remaining, the Braw Lads should really have gone beyond the 200 mark.
In reply, Dunnikier gave Gala a number of chances which they failed to take, with Robbie Irvine the main victim as his superb bowling went unrewarded.
Dinesh Tharanga stepped up to the mark to take two wickets but Aziz and Rafi put on a 78-run partnership to set up what seemed like a Dunnikier win.
But, to their credit, Gala never gave in, with man of the match Rhode (3 for 27), Pringle (1 for 22) and Josh Irvine (2 for 33) taking regular wickets to leave Dunnikier 167 for 9.
There was controversy at the end as number 11 batsman Ali requested a runner as he walked to the crease with an injury, which Gala argued was not received during the game, but the request was allowed.
There was more drama in the last over as Gala celebrated what they thought was a run out, only for the Dunnikier player filling in at square leg umpire to claim he had not given his team-mate out.
Needing two to win off the last ball, Dunnikier then scrambled one run to seal the draw.
J. Halls ct F. Rafi b M. Safi 2
R. Irvine ct Hamidoullal b S. Rowley 24
J. Irvine ct M. Akhtar b M. Sarfraz 14
D. Millar run out 6
A. Rhode lbw M. Sharif 37
K. Gallagher lbw S. Rowley 10
F. Rutherford ct F. Rafi b S. Rowley 5
D. Tharanga b M. Safi 7
G. Mein ct F. Rafi b M. Sharif 20
R. Pringle b M. Safi 14
A. Swanson not out 0
Extras – 38
Total – 177 for ten wickets
M. Safi 8.3-0-31-3; M. Sarfraz 5-0-36-1; M. Akhtar 10-2-18-0; S. Rowley 10-3-30-3; M. Sharif 5-0-34-2; Z. Hamidoullal 1-0-11-0.
M. Akhtar b D. Tharanga 29
S. Rowley b D. Tharanga 7
T. Aziz ct K. Gallagher b A. Rhode 44
F. Rafi ct D. Millar b R. Irvine 30
M. Sharif lbw J. Irvine 19
M. Sarfraz ct sub (J. Crooks) b A. Rhode 8
S. Saeed lbw J. Irvine 1
M. Arman ct D. Millar b A. Rhode 0
M. Safi not out 17
Z. Hamidoullal b R. Pringle 4
A. Ali not out 2
Extras – 16
Total – 177 for nine wickets
R. Irvine 10-1-47-1; D. Tharanga 10-2-22-2; A. Rhode 10-3-27-3; G. Mein 4-0-23-0; R. Pringle 6-1-22-1; J. Irvine 10-1-33-2.