After Berrington won the toss, opener Charlie Tear played well for his 68 after George Munsey and Brandon McMullen both fell victim to the new ball.
His contribution, paired with his captain's fine effort, allowed Leask to attack in the final 10 overs.
Scotland's number seven struck 12 fours and three sixes on his home ground in a brutal display of power hitting.
Berrington's side then started in fine fashion with the ball, reducing Nepal to 34-3, including a first international wicket for Jones.
Nepal rebuilt through Bhim Sharki (73) and Aarif Sheikh (39), but when they were removed by Leask and Jones, respectively, Scotland appeared to be in total control.
However, a partnership of 105 for the seventh wicket between Dipendra Singh Airee (56) and Sompal Kami (67) brought Nepal right back into the contest.
Jones caught Singh Airee off Charlie Cassell's bowling and removed Kami himself as the ball followed the Essex seamer on his first Scotland outing.
Nepal needed seven to win off the final over - bowled by Safyaan Sharif - and Jones was called into action again, taking a towering catch at long-leg to remove the dangerous Karan KC.
It left Nepal needing five off three balls, but Sharif held his nerve to bowl Lalit Rajbanshi and win the game for Scotland.
It was the seamer's 258th wicket for Scotland, taking him level with Majid Haq as his country's all-time leading wicket-taker.