The 2015 champion won the first 11 games as she homed in on a double-bagel (6-0 6-0) victory before the big-hitting Vekic finally established a foothold in the match.
Vekic reeled off four successive games and came within two points of levelling up the second set before Watson broke for a sixth time to seal the contest in a little more than an hour.
"She's a big hitter, who can hit winners from pretty much anywhere," Watson, who converted all but two of her eight break-point opportunities, told reporters.
"When that started to happen, the most important thing was not to panic. When you go that far up, it can feel a bit up and down when you lose the next four games, but I had to realise that I was still 5-4 up and playing well."
Meanwhile, the repeat of last year's title showdown failed to materialise after losing 2017 finalist Monica Niculescu of Romania pulled out with a neck injury, allowing Belgium's Mertens to advance to the last four without hitting a ball.
Elsewhere, fifth seed Lesia Tsurenko ended Belarusian Aryna Sabalenka's strong run with a 6-1 6-1 triumph to also secure a place in the last four.
The Ukrainian advances to meet Romania's Mihaela Buzarnescu, who secured the last semi-final berth with a 7-6(5) 6-1 win over American Alison Riske.
(Reporting by Amlan Chakraborty in New Delhi; Editing by John O'Brien)
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