Top seed Medvedev, runner-up to Djokovic at the Australian Open in 2020 and Rafael Nadal last year, also had a comfortable victory, beating fellow Russian Karen Khachanov 6-3 6-3.
Djokovic, who leads the head-to-head 8-4, said: “I’m happy with the way I’m playing, the way I’m feeling on the court. I know that a grand slam is around the corner but I really want to focus my attention here.
“Medvedev, maybe I cannot get tougher than that at the moment. I look forward to it. It’s going to be a great semi-finals for sure.
“But, at the same time also, I think the biggest test so far for both me and him, I think this is what we want coming into Melbourne and the Australian Open.”
The other semi-final features American Sebastian Korda – who upset sixth seed Jannik Sinner 7-5 6-1 – and Japan’s Yoshihito Nishioka.
In the women’s event, top seed Ons Jabeur battled to a 7-6 (5) 7-5 victory over Ukrainian Marta Kostyuk and will next take on 18-year-old Czech qualifier Linda Noskova, who continued her excellent week with a 6-4 6-7 (3) 7-6 (6) win against Victoria Azarenka.
Noskova, ranked 102, said on wtatennis.com: “I definitely didn’t expect to play semi-finals with wins over (Daria) Kasatkina or Azarenka. I just came here for qualies. I don’t know what I’m doing here.”
Second seed Aryna Sabalenka beat Marketa Vondrousova 6-3 7-5 and next faces Romanian Irina-Camelia Begu.
Rain continued to blight the ASB Classic in Auckland, with matches again played indoors. Coco Gauff defeated Zhu Lin 6-3 6-2 in the quarter-finals but third seed Leylah Fernandez was beaten by Ysaline Bonaventure.