The halt in proceedings came 63 minutes into the opening set, after Wawrinka saved a Nadal set point for 5-all, with the Spaniard then holding serve for 6-5.
When the weather passed, the pair returned, with Nadal wrapping up the set on his second chance.
The second set was a battle as the Swiss, a three-time Grand Slam champion, fought back from an early break down, took a 2-1 lead, with the pair again trading breaks in the fifth and tenth games.
Nadal came from a mini-break down in the tiebreaker and secured victory on his second match point.
"It was a good match, a very positive victory for me over a tough opponent," Nadal said.
"I'm happy to see Stan playing well again. We had a good quality of tennis.
"I'm very pleased, I needed a match like this. It does much for the confidence."
Nadal's win was his 17th from 20 played against Wawrinka.
Second seed Alexander Zverev, the holder, lined up a re-run of last week's Washington semi-final as he again takes on Stefanos Tsitsipas.
Zverev rolled over Russian Daniil Medvedev 6-3, 6-2 in 52 minutes while never facing a break point.
Tsitsipas called his knockout of Djokovic, "the best match of my career. I knew I was playing pretty good today."
The number 27 will play the first Masters 1000 quarter-final of his career.
"He played very well and deserved to win without a doubt," Djokovic said. "I just played not that great.
"I didn't return well. It wasn't that great of a match."
Bulgarian fifth seed Grigor Dimitrov laboured for almost two and a half hours to subdue Frances Tiafoe of the United States 7-6 (7/1), 3-6, 7-6 (7/4).
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