The 23-year-old from Stockport, whose sister, Naomi, is the British No. 3, beat the Latvian 6-3 6-0 after having to get through two rounds of qualifying.
Broady has won a match at Wimbledon in 2015 but had never tasted victory at the elite level of the men's tour until his meeting with world No. 212 Gulbis, who beat Juan Martin del Potro on his way to the third round of this year's Wimbledon.
He took the first set with a solitary break of serve and then blew his opponent away in the second on his way to a 'bagel.' Broady, ranked 242 in the world, will play fourth seed Adrian Mannarino in the second round.
Elsewhere in the draw, Mikhail Youzhny saw off John-Patrick Smith 7-6 (8-6) 6-4, while Lithuanian Ricardas Berankis recovered to beat Daniel Masur 3-6 6-3 6-2 before seventh seed Viktor Troicki beat Andrey Kuznetsov 7-6 (8-6) 7-6 (7-2).