"We got up a break and kind of relaxed and they played some good points to get back on serve," said Venus. "It was great to be able to get through that first set and then we started to play a bit better in the second and managed to finish it off."
Venus and Klaasen are aiming to take the extra step this year, after being eliminated at the semifinal stage in 2018.
Venus is a previous doubles champion in Auckland (2016 with Mate Pavic) and also reached the last four in 2012 alongside compatriot Daniel King-Turner.
Venus and Klaasen will play the Bryan brothers in the semifinals, with the famous American pair beating Kiwi Artem Sitak and partner Austin Krajicek 6-7, 6-2, 2-10 in an entertaining match on centre court.
It was the one that got away for Sitak and Krajicek. Neither player was broken across the entire match, and they seemed to have the momentum after the second set.
But the Bryans were cool and clinical, showing all of the experience garnered over almost two decades on court together.
The final match on Thursday night didn't start until 11:15pm, and was played on an outside court.
Daniell and Wesley Koolhof were edged 7-6 (4) 7-6 (5) in 87 minutes by Jan-Lennard Struff (Germany) and Ben McLachan (Japan) , in front of a vocal crowd, with equal amounts of support for Daniell and former Kiwi McLachlan.
It was a marathon effort from Struff, who had just hours before beaten Pablo Carreno Busta in a thrilling three-hour singles quarterfinal, 7-6, 6-7, 7-6.
He was a constant menace, both on serve and with his thunderous returns.
But it was an extremely tight match. Struff and McLachlan didn't face a single break point, while Daniell and Koolhof saved the break opportunities created on their serves.
But they faded slightly in both tiebreaks, allowing their opponents to take the initiative.
The pairing now go up against Oliver Marach and Pavic in a repeat of the 2018 Australian Open semifinal.
Daniell, Stiak and their respective partners head to Melbourne to begin preparations for the Australian Open next week.
