The Dutchman avenged his defeat to Wade at the same stage of last year's tournament and will face Gerwyn Price on Saturday, following the Welshman's 16-12 win over Ross Smith in the opening quarter-final.
Van Veen was below the high standards he's shown previously, but produced big moments when they mattered most at the Winter Gardens.
After a scrappy start, he broke clear with a 160 checkout to move 9-6 ahead before extending his lead to 11-7.
In response, Wade landed finishes of 79 and 121 to win four of the next five legs and go level at 11-11, before another 121 kept his hopes alive late on.
Van Veen hit a 118 checkout to edge 14-12 in front and won the final two legs to seal victory.
"My scoring has been okay the whole tournament, but today it just wasn't up to par. My finishing got me through that. It was a tight battle," Van Veen said.
"Even though I'm not scoring as well as I want to, I'm still getting through to the semi-finals. I've got about 20 hours to put that right.
"Hopefully tomorrow I'll be a lot better because I'll need to be against Gezzy.
"This is the second biggest major after the World Championship, so I'm so glad to get into the semi-finals for the first time."
Earlier, 'Gezzy' Price reached the last four for the second time after producing his best display of the tournament so far to beat Smith.
The Welshman averaged 102.78 and hit 10 maximums as he built a 12-8 lead by edging out each of the opening four sessions 3-2.
Smith responded by reeling off four straight legs with finishes of 68, 72 and 80 to draw level at 12-12.
Price regained control, taking out 121 on the bull before adding a 91 finish to win the final four legs and book his place in the semi-finals.
"Ross played a good game, and it's nice to play well when someone is playing well against you. It gives you confidence," Price said.
"When Ross got it back to 12-12, I thought: 'Here we go again', but thankfully I rolled off four legs on the bounce, so happy days.
"I'm trying to keep myself calm. It worked today, but on other occasions I need to let my emotions go. It has to come naturally.
"Like I said, I'm going to get better and better as this tournament goes on, and hopefully in the next round I'll play better again."
Saturday's semi-finals begin with Price taking on Van Veen before defending champion Luke Littler meets Dirk van Duijvenbode for a place in Sunday's final.
Results
Gerwyn Price 16-12 Ross Smith
Gian van Veen 16-13 James Wade
