Welsh team captain Davies, who competes in the F42 category, took an early lead in the second round with 46.83m which scored him 1,012 points.
But Greaves, who is in the F44 category for athletes with less severe impairments, responded in the third round with 59.21 and 1,023 points.
“I wanted to put on a great performance out there,” said Greaves, 31.
“I knew I was in good shape and I can perform at the right time.
“We’ve competed against each other all season and it has been a battle and I knew Aled could have have popped one out but when I threw 59m I thought I had a good chance of winning.”
“I’m trying to look at the positives but I’m a bit gutted,” Davies said.
“It’s not the colour I wanted. But I’m sure in time I will appreciate it.
“I know I could have thrown a bit further but it’s just what happens in these major championships.”
Greaves, who along with Davies will be competing in the IPC Athletics European Championships in Swansea in August, paid tribute to the Hampden Park crowd.
“This atmosphere is on a par with London. I wanted to put on a great show for the crowd and they pushed me all the way and I am thankful for the support.”