Cross moved 3-0 up and responded to a Dobey 14-darter with two of his own and a 12-dart leg as he moved into a 6-1 advantage.
Dobey finished 145 to hit back in leg eight, but Cross fired in six perfect darts as he restored his advantage before pulling away to seal victory with a 130 checkout.
SEE ALSO Rob Cross in the 2019 Betfred World Matchplay: All you need to know | Pundits purr over world champion Cross | Rob Cross seeded second in 2019 Betfred World Matchplay draw"It's nice to get over the winning line with no fuss," said Cross. "I wasn't great in the first five legs, and even though I was 4-1 up I was a bit edgy, but I settled down after that and took the initiative in the second session and started to play proper darts.
"If Chris had won the first leg, when he missed a couple of doubles, it could have been a different game. He's a class act and he's one for the future.
"I think it's the first time in three years here that I've found my feet and pushed on like that.
"I probably get better as the game goes on, and overall I'm happy with how I'm playing - but I don't feel like I've played my best darts yet. I love this event and I want to do well this week."
The first round continues on Sunday with a double session at the Winter Gardens, which includes two-time champion Michael van Gerwen up against Steve Beaton and 2007 winner James Wade taking on 2018 semi-finalist Jeffrey de Zwaan.
Two-time World Champion Adrian Lewis takes on three-time Lakeside Champion Glen Durrant in an eagerly-awaited tie, while 2018 finalist Suljovic plays Jermaine Wattimena.
The afternoon session opens as Darren Webster takes on Polish debutant Krzysztof Ratajski, before Dave Chisnall meets Max Hopp, Ian White takes on Joe Cullen and Michael Smith clashes with debutant Jamie Hughes.
2019 Betfred World Matchplay
Saturday July 20 First Round
Mervyn King 10-5 Nathan Aspinall
Stephen Bunting 13-12 Gerwyn Price
Gary Anderson 10-6 Danny Noppert
Rob Cross 10-3 Chris Dobey
Sunday July 21 (1300 BST) First Round
Darren Webster v Krzysztof Ratajski
Dave Chisnall v Max Hopp
Ian White v Joe Cullen
Michael Smith v Jamie Hughes
Evening Session (1930 BST) First Round
James Wade v Jeffrey de Zwaan
Mensur Suljovic v Jermaine Wattimena
Michael van Gerwen v Steve Beaton
Adrian Lewis v Glen Durrant
Format Best of 19 legs: Each game must be won by two clear legs, with up to a maximum of five additional legs being played before the sixth additional leg is sudden-death. For example, should a First Round game (best of 19 legs) reach 12-12, then the 25th leg would be the final and deciding leg.