The Northern Irishman won by three shots after shooting a level-par 70 to win his first major title.
Afterwards he paid tribute to Heather, who died in 2006 from breast cancer, and their sons Tyrone and Conor.
Clarke said: "There's obviously somebody watching from up there and I know she'd be very proud. But I think she'd be more proud of my two boys."
"It's for the kids," he added. "They played golf at Royal Portrush this morning and were watching on TV.
"Heather would probably be saying, 'I told you so'."
The 42-year-old becomes Northern Ireland's third major champion in little over a year, following Graeme McDowell and Rory McIlroy's wins at the US Open.
Clarke's five-under par total for the tournament saw off the American pair of Phil Mickelson and Dustin Johnson, who finished joint second, with Denmark's Thomas Bjorn a shot further back in fourth.
It left the Northern Irishman promising "a long and very enjoyable" night of celebration.
"I'm on Weight Watchers tomorrow morning," he quipped after completing his victory. "There's five points in a pint of Guinness... I think this could probably be a bad week for me to try and start."
Clarke will next be in action at the Irish Open in Killarney on 28 July and he promised: "I may not be sober for the Irish Open, but I will be in Killarney.
"It's pretty amazing right now, to tell you the truth. It's been a dream since I've been a kid to win the Open, like any kid's dream is, and I'm able to do it, which just feels incredible.
"I played okay today, I did what I needed to do. The last couple of holes I was trying not to make any stupid mistakes. I just tried to play really carefully and it was good enough to win."
The Sandwich tournament was the 54th major in which Clarke had participated, and he had not had a top-10 finish for a decade.
But he said: "I've got here in the end. It may be the only major that I win, it may not be the only major that I win, but at least I've gone out there today and did my best, and my best was good enough to win.
"If it hadn't come off and I hadn't won, I could still have said I did my best. I ask my two boys to do their best and that's what they do, so I think their dad should try and do the same."
Clarke went into the final round with a one-shot lead over playing partner Dustin Johnson and five ahead of Mickelson, but that gap was wiped out in seven holes as left-handed Mickelson registered three birdies and an eagle.
But the Northern Irishman refused to panic and matched Mickelson's eagle on the seventh before chalking up nine pars in a row to give him breathing room over the last few holes.
Mickelson and Clarke have long been good friends and they shared a private moment after Clarke had sunk his winning putt.
The pair became close because of their shared experience of watching loved ones battling breast cancer - Mickelson's wife Amy and mother Mary have both experienced it - and Clarke said: "It is something that I could talk about but I'm not going to.
"(But) he has turned into a very good friend of mine through thick and thin, and he said some very, very kind words to me there which is great."
Mickelson said his friend would be a hugely popular Open champion. "He couldn't be a nicer guy," he said. "There's a lot of players that are extremely happy for him.
"He was one of the first people that called us, Amy and I, a couple years ago. He's been through this and couldn't have been a better person to talk to.
"He's a tremendous person and a very good friend, and I couldn't be happier for him."
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