Shaheen Afridi (4-18) had all but won the game for Lahore at their home ground in Gaddafi Stadium, with Islamabad 163-9, before Musa smashed two boundaries off the leftarm fast bowler to reduce the target to nine runs off the last over.
Musa (17 not out) smacked another leftarm fast bowler Usman Shinwari over long on for a six off the first ball in the last over before running the required three runs over the next four balls as Islamabad achieved the target of 183 with a ball and a wicket to spare.
Put into bat,experienced allrounder Mohammad Hafeez missed out on a deserved century but his unbeaten 98 off 57 balls propelled Lahore's total to 182-6. Hafeez hit seven sixes and seven fours. Islamabad captain Shadab Khan (2-14) and Faheem Ashraf (2-42) did much of the damage.
Islamabad recovered in its chase through the first PSL half century for Shadab (52) after losing overseas batsmen Colin Munro (2) and Luke Ronchi (1) in the first two overs - bowled by Afridi and Hafeez.
Colin Ingram hit a 14-ball 30, with three sixes and one four in left-arm spinner Samit Patel's over which kept Islamabad in the hunt.
But Afridi returned and took three wickets in his last two overs before Musa's brilliance with the bat denied Lahore victory.
Islamabad now has won two of its three matches and has four points, while Lahore lost both its home games — against Multan Sultans and Islamabad.
At Karachi, 21-year-old Azam Khan continued to inspire with his batting as Quetta also notched its second victory in the tournament
The 21-year-old Azam, son of former Pakistan captain Moin Khan who is now coach of Quetta, scored 46 off 30 balls and guided the team to 157-5 with an over to spare.
Quetta now has four points after playing three games in four days. It beat two-time champion Islamabad United in the opening game before losing to 2017 champion Peshawar Zalmi on Saturday.
Sarfaraz remained unbeaten on 37 off 28 balls and featured in a match-winning 85-run fourth wicket stand with Azam after Quetta's overseas signings Shane Watson (27) of Australia and Jason Roy (17) of England were run out.
Azam, who scored 59 in Quetta's victory in the opening game against Islamabad, missed his second half century when he was run out in a mix-up with Sarfaraz in the 18th over, but by that time Quetta were well on their way to victory.
Earlier, Karachi struggled to put on partnerships and was restricted to 156-9 after captain Imad Wasim won the toss and elected to bat.
Pakistan's star batsman Babar Azam (26) survived a controversial caught-behind appeal on 24 against young pace sensation Naseem Shah due to a technology failure.
Quetta went for a television referral, but the ultra-edge technology didn't work and the third umpire retained the on-field umpire's not out decision.
However, Babar didn't last much longer when he dragged Tymal Mills' full length delivery back onto his stumps and was clean bowled.
Alex Hales, playing his first game in this year's PSL, top scored with 29 and Iftikhar Ahmed scored 25 before both fell to Pakistan international fast bowler Mohammad Hasnain, who took 3-33.
Karachi, which has never won the PSL, has two points after two games. It won a high-scoring opening match against Peshawar Zalmi on Friday.
Pakistan's premier domestic Twenty20 tournament, which is being played in the country for the first time in its entirety after four years, resumes Wednesday.