Draper has always been deemed to have the talent to go far in the game, but his progress has been hampered by injury and fitness issues.
A shoulder problem ruled him out of the British grass-court season last year, leaving him ranked 138th in the world when he returned in August.
“Winning a title helps me to believe more and more,” said Draper, who lost ATP finals in Sofia and Adelaide before his Stuttgart success.
“It’s been tough, I have been close a few times and not quite getting it over the line.
“I think it really changes my belief system a little bit as well as to what I can achieve.”
Draper said he had “little time” for his achievement to sink in, having flown back to London shortly after beating Berrettini and practising at Queen’s on Monday.
He will play for the first time on home soil as the British number one when he faces Argentina’s Mariano Navone in the first round on Tuesday.
“Jack, I think, has bigger goals to aspire to than being British number one,” said former world number one Murray.
“In my opinion, he can get right up to the top of the game, right to the top of the world rankings.”
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