Britain's world number 11, who lost in the final to Stefanos Tsitsipas in 2021, had match points in the second set before losing it in a tie-break.
But he broke early in the decider and Molcan's challenge faded.
"This one feels the best so far, I don't know why, it's so special," Norrie, 26, told Amazon Prime.
"It's my first one on clay. It means a lot for me - I couldn't be happier with how I handled everything this week."
Norrie, whose four ATP titles have come in the space of just 10 months, found himself a double break up in the first set before a break in play for medical attention to Molcan, who had cut his knee, saw the world number 47 start to get a toehold in the match and pull back a break.
Norrie held off the fightback and broke his opponent's serve again to close out the first set 6-3 but the second was a much tighter affair.
At 5-4 up, Norrie earned two championship points on the Molcan serve but the 24-year-old levelled the match before holding on for a tie-break which he won comfortably.
But once Norrie secured the first break of the third set, Molcan - still searching for a first title after three finals - faltered and the Briton ran through the set to complete victory.
Norrie will play his first-round match at the French Open against France's Manuel Guinard on Monday.
Molcan faces Federico Coria of Argentina at Roland Garros.