The match had ground to a halt a day earlier after Medvedev cruised through the opening two sets and surrendered the third, but the Russian returned to the roofless Court Two on another soggy day and held his nerve to prevail.
A backhand winner on matchpoint in the fourth-set tiebreak settled the clash, as Medvedev booked a meeting with Bulgarian 10th seed Grigor Dimitrov.
"I knew it's going to be a long day. I actually thought we wouldn't play at all yesterday. But the weather was better than what it was supposed to be. So we managed to get full three sets. Then we got unlucky with the rain," Medvedev said.
"It definitely wasn't easy, but I'm happy to win. I feel good physically, which is important.
"I'm looking forward to tomorrow."
Medvedev made a blistering start to the match winning the opening five games before Struff got on the board thanks to an unforced error, as the fifth-seeded Russian easily closed out the first set with a fiery ace.
World number 41 Struff's chances of a second victory over Medvedev in their seventh tour-level meeting faded further after he lost three straight games towards the end of the second set, having gone toe to toe with his opponent.
With dark clouds hovering in the air, Struff rekindled his hopes by winning a see-sawing third set where Medvedev's level suddenly dipped, before rain stopped two games into the fourth and frustrated the duo again on a stop-start Saturday.
Medvedev was delighted with his level against Struff, who can be a challenging opponent on the sport's slickest surface with his powerful hitting.
"Right now, I feel I'm playing well. I beat a very strong grasscourt opponent," Medvedev said.
"Actually, even one set 6-1, which I was surprised. I'm still surprised against Struff. I'm happy with my level.
"The only way to win the whole thing is to play better and better every match. So far I've managed to do it."
