Chasing 277 for victory at Titwood, the Saltires were staring at disaster against India's star-studded second string.
They had slumped to 64-7 and looked in danger of being completely humiliated.
Instead Berrington and Moneeb Iqbal staged a thrilling fightback as the Scots carved out a sensational one-wicket win with four balls to spare.
Greenock star Berrington notched his first international century while Iqbal lent crucial support with 67.
Berrington was forced to dig in as the Indians initially took control but he opened out to reach the landmark score from 106 balls, hitting eight boundaries and four sixes. He was eventually out with 39 still needed, but West of Scotland's Iqbal took the Scots to the brink of victory before being run out while Gordon Goudie sealed glory with 26no from only 20 balls.
Dhaval Kulkarni and Sudeep Tyagi had helped themselves to three wickets apiece as Scotland's top order caved-in.
New skipper Gordon Drummond joined the collapse before Berrington and Iqbal's 174-run stand changed the game.
Home hopes were raised at the start of the day when Goudie picked up two wickets in his first over to reduce the Indians to 11-2. The West of Scotland seamer produced a brute of a delivery to take out Shikar Dhawan's middle stump and then had skipper Chet Pujara caught in the slips.
IPL star Ajinkya Rahane led the revival with a century while Manish Pandey and Manor Tiwary lent good support.
Pandey hit 32 before he became the first of Berrington's three victims while Tiwary helped Rahane add 138 for the fourth wicket.
Mumbai Indians player Rahane hit 12 boundaries in his 108 as the visitors looked on course for 300-plus, but the Scottish bowlers held their rivals to 276-9 with Goudie picking up 4-46 and Berrington 3-45.