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With India only able to claim the wicket of Clarke (128) in the final over before bad light stopped play, Australia finished the second day on 517-7, with Smith on 162 not out and Mitchell Johnson having yet to score.
After losing three wickets in the last five overs on the opening day to slump to 354-6, Smith and Clarke's partnership put the home side on top in the first of four Tests against India.
On a rain-affected day at the Adelaide Oval, Clarke surprised many by walking out to bat when play got underway, while the skipper impressed everyone with his brave effort to fight through the pain of his back injury to reach three figures.
Clarke had retired hurt on 60 just before tea on Tuesday after twisting his troublesome back when evading a short-pitched delivery.
In a shortened first session, Clarke lacked his normally textbook footwork, standing as upright as possible in order to take strain off his back and seemingly relying on his hand-eye coordination to score.
It did not stop him from hitting a boundary on the first ball of the second over he faced, however, with Clarke reaching 85 when rain next stopped play, while Smith was also flying at 98 not out.
After an extended break for lunch, Smith dispatched Varun Aaron's first delivery for two runs to notch his century.
Having embraced Clarke, the 25-year-old walked over to the 408 painted on the Adelaide Oval outfield in memory of former Australia batsman Phillip Hughes, pointing at the number with his bat before looking to the sky.
Less than two overs were completed before rain again forced the players off the ground with Clarke on 98.
The contest resumed almost two hours later and Clarke needed just five more balls to reach his seventh Test century at the Adelaide Oval - more than any other player and from just 10 matches.
With both having reached triple figures, Smith and Clarke appeared to relax and hit eight boundaries in six overs before, frustratingly, rain fell again.
India had four chances to end Smith's innings late in the day with Wriddhiman Saha missing a stumping before the final rain delay, while Cheteshwar Pujara, Virat Kohli and Ishant Sharma dropped him afterwards.
Instead it was Clarke who fell, dismissed by Karn Sharma (2-143) late in the day when he swept straight to Pujara.