Leg-spinner Yasir Shah grabbed a career-best seven for 76 to skittle Sri Lanka for a paltry 206 in their second innings just before tea on the fifth and final day.
Pakistan set a victory target of 90, and raced home in 11.2 overs with Mohammad Hafeez remaining unbeaten on 46 and Ahmed Shehzad on 43.
The second Test starts at the P. Sara Oval in Colombo on Thursday.