Nadal commented "I probably played a little bit worse today than yesterday", adding "but in general terms [it's] been a positive victory again".
The top-seeded Serb, who had not dropped a set in four matches since the round of 16 at the Australian Open in January, looked out-of-sorts throughout the 98-minute contest.
The multiple Grand Slam champions both won their quarter-finals on Friday, although Nadal in particular had to work hard to overcome Russian Karen Khachanov 7-6 (2) 7-6 (2) in a match during which he required treatment on the knee.
Federer, meanwhile, broke Pole Hubert Hurkacz once in each set for a 6-4 6-4 win in the first career meeting between the pair and was perhaps more bothered by his early start.
It took Roger Federer just over an hour to storm past 23rd seed Kyle Edmund of Britain 6-1, 6-4, moving closer to his sixth Indian Wells title.
"Amazing is an understatement", said Andreescu, who is playing her first season as a professional.
Nadal, who lifted the Indian Wells trophy in 2007, 2008 and 2013, is in the last eight for the first time in three years after missing last year's edition, skipping both Indian Wells and the Miami Masters with a hip injury.
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"I lot has happened in the past 18 months", he said.
Born in Canada, Andreescu first took up tennis at age 7 in Romania, where her parents worked. Hurkacz, 22, will be playing in his first ATP Masters 1000 quarter-final. "They warned me they were going to put me on at 11.00 (a.m.)".
"It will be amusing that somebody other than Novak is still in", he said of his famous countryman.
Thiem booked a meeting with 18th-seeded Gael Monfils of France, who defeated Philipp Kohlschreiber 6-0, 6-2 in 57 minutes a day after the German stunned world number one Novak Djokovic in the third round.
"I was relieved I got through that one", Federer told reporters after defeating Edmund.
2016 finalist Milos Raonic kept his hopes of returning to a fourth Indian Wells semifinal by upending Jan-Lennard Struff 6-4, 6-3.
The two greats' comfortable wins came after the top half of the draw was rocked by the upsets of top-ranked Novak Djokovic and No.3 seed Alexander Zverev on Tuesday.