Shelton's 6-2, 7-5 win over Flavio Cobolli also made him the fifth American this century to bag a clay-court title outside the United States, joining Agassi, Andy Roddick, Sam Querrey and Sebastian Korda.
The 23-year-old said the "huge" triumph underlined his ambitions before the French Open, which begins on May 24.
"Moving forward I have big ambitions for the clay courts, a surface I want to get better on each year. It's become one of my favourite surfaces to play on," Shelton said.
While the American women have had plenty of success on the sport's slowest surface, with Coco Gauff winning the French Open crown last year, the attention will now turn to whether the US men can leave their own mark in Paris.
With Tommy Paul and Frances Tiafoe making the Roland Garros quarter-finals last year, Shelton said things were looking up as American men aim to end a Grand Slam drought going back to 2003 when Roddick won the hard-court US Open.
"Success on clay is coming back," he added.
"I'm looking forward to being part of this progression of US men's tennis on clay.
"On the women's side, they have a lockdown as they won the French Open last year. We as men have some more to do but we're heading in the right direction."