Rest of India were 328 for two at stumps, with Vijay and S Badrinath (1 not out) at the crease, at the Chinnaswamy stadium.
Rajasthan were all out for 253 on the first day.
Vijay hit 18 fours and five sixes during his 257-ball innings, which turned out to be his tenth first-class century.
Ajinkya Rahane and Cheteshwar Pujara contributed 81 and 78 respectively, and both were involved in big century stands with the Tamil Nadu opener.
After adding 173 runs for the first wicket with Rahane, Vijay was joined by Pujara, and the two raised 153 runs for the third wicket to put their team in a commanding position, a day after the bowlers set it up with Umesh Yadav leading the way with a five-for.
Vijay notched up his century by ambling across for a single off Madhur Khatri, a fleeting moment after hitting him out of the park for a maximum.
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Out of the Indian team, Vijay used the opportunity with a well-stitched innings, that should force the selectors to take note.
Just before the close of play, Vijay reached 150 after he pushed Khatri for a single to long off.
The first-wicket partnership was broken when Rahane, who reached 5,000 runs in first-class cricket, was run out after facing 143 balls. His knock was studded with twelve hits to the fence.
The second wicket fell in the form of Pujara, who was snapped up by Dishant Yagnik off Aniket Choudhary for a well-made 78 off 140 balls, and laced with eleven boundaries and a six. MORE