The Netherlands paceman led the way with his four-wicket haul.
Surrey slumped to 93 all out with only the Curran brothers, Sam and Tom, reaching 20.
Craig Meschede and Dean Cosker claimed two wickets apiece as Surrey fell apart and Glamorgan strolled to an eight-wicket win with 46 balls to spare.
Glamorgan's new opener David Lloyd hit 31 and captain Jacques Rudolph was 34 not out.
Surrey chose to bat but their opening stand of 28 was the highest partnership of the innings, with seven wickets then falling for 32.
They were further hindered by veteran Pakistan all-rounder Azhar Mahmood retiring hurt with a calf injury without scoring.
Glamorgan's next match is also in the T20 Blast, at home to Essex on 1 June, while Surrey travel to Somerset in the Championship on 28 May.
Man of the match Van der Gugten told BBC Wales Sport: "I thought as a team we bowled really well, we kept them to a low total and our batters just did the job very professionally.
"A couple of wickets went my way, anyone could have taken a bagful but luckily it was my night. [South Africa pace bowler] Dale Steyn coming will strengthen our group and we're looking forward to working with him."
Glamorgan captain Jacques Rudolph told BBC Wales Sport: "The last two years we've come down here we've had very good performances, as far as clinical performances that's as good as you're going to get. Our bowling was outstanding and Timm van der Gugten again was unbelievable and we backed it up well with the batting.
"Surrey's top three batters can be very dangerous, I continuously tried to attack and get wickets, and it's nice to get some confidence in the dressing room.
"As far as I'm aware Dale Steyn [currently in India] is very keen to play on Wednesday (against Essex), Timm van der Gugten has shown with a bit of extra pace the kind of X-factor we need in our team. With him and Dale in the same team, we can be really excited about that."
Glamorgan won for the first time since July 2015 with Timm van der Gugten taking four wickets as they thrashed Surrey in their opening T20 Blast game.