In a rematch of last year's final at the Caja Magica, Alcaraz snapped Struff's six-match winning streak with a 6-3, 6-7(5), 7-6(4) result to squeeze through to a last-eight clash with seventh seed Andrey Rublev.
Top-seeded Jannik Sinner has been dealing with a hip problem but managed to overcome Karen Khachanov 5-7, 6-3, 6-3 to reach his fourth Masters 1000 quarter-final in as many events this season.
Alcaraz, who missed Monte Carlo and Barcelona with a forearm injury, saved eight out of 10 break points and saw four match points come and go before he completed the two-hour 52-minute win.
"How much I've missed this! Quarters next," wrote the former world number one on the camera lens after the match.
Alcaraz, who is now 24-0 on Spanish clay since the start of the 2022 season, is bidding to become the first player to win three consecutive Madrid titles.
After blowing a 2-0 lead in the second set, Alcaraz opened up a 3-1 advantage in the tiebreak, only to lose the next five points and go on to surrender the set to Struff, who came into the tournament on the back of a maiden ATP title run in Munich.
A break of serve in the fourth game of the decider gave Alcaraz the score cushion he needed but the two-time Grand Slam champion was broken while serving for the victory from 40-0 up at 5-3.
Struff saved four match points and the contest fittingly went to a deciding tiebreak. Alcaraz let a 3-0 lead slip, but he got his hands on more match points by landing an inch-perfect lob and he closed on his fifth opportunity of the match.
"This match reminds me of last year's. It was a great fight," said Alcaraz after matching Rafael Nadal's record winning streak in Madrid.
"It was difficult for me to deal with my emotions, to handle difficult moments. Serving for the match, it was really difficult for me to get broken after being 40-0 up, but I'm really happy that at the end it didn't affect me and my mentality and I kept fighting."
Sinner won 80 percent of points behind his first serve against Khachanov, hitting 35 winners to 29 unforced errors in the two-hour nine-minute battle.
The reigning Australian Open champion will face Felix Auger-Aliassime for a place in the semi-finals.
"For sure I'm not physically at 100 percent today. Tomorrow I have one day off, which can help me to get through," said the Italian.
Meanwhile, third seed Daniil Medvedev completed his full set of Masters 1000 quarter-finals reached by defeating Alexander Bublik 7-6(3), 6-4 to book a meeting with Rafael Nadal or Jiri Lehecka.
After grabbing the opening set tiebreak, Medvedev cruised to a 5-1 lead before Bublik halted his momentum and narrowed his deficit to just one game.
But Medvedev had a second chance to serve for the victory and this time he comfortably converted his first match point to make it to the last-eight stage or better at each of the nine Masters 1000 events.