Torrential rain had already washed away 34 overs on day one and the entire second day's play and India lost six wickets in their bid for quick runs in two stop-start sessions before the entire third session was called off.
Ravichandran Ashwin (two) and Harbhajan Singh (seven) were at the crease when rained forced the players off the ground, virtually killing the prospect of any result in the match.
After India resumed on 239 for no loss, Shakib (4-105) and leg-spinner Jubair Hossain (2-113) took all six wickets between them as the visitors went for quick runs.
Shakib ended India's 283-run opening stand by dismissing Shikhar Dhawan (173) and also claimed the wickets of Vijay (150), Rohit Sharma (six) and Rahane (98).
Vijay steered left-arm spinner Taijul Islam for a four to bring up his sixth test hundred in the morning session but his opening partner Dhawan could not better his personal test best of 187.
The southpaw gave Shakib a soft return catch to depart after a 195-ball knock that included 23 boundaries.
Shakib struck again in his next over, uprooting the off-stump of Rohit, the only batsman with two 200-plus scores in one-dayers who continues to struggle in tests.
India skipper Virat Kohli (14) was slightly unlucky as his fluffed drive resulted in an inside edge which hit his pad before rolling onto the stumps to knock off the bail.
Vijay and Rahane, however, had little problem negotiating the hosts' limited bowling resources and they added 114 runs for the fourth wicket.
Vijay was trapped leg-before by Shakib after a 272-ball knock that included 12 boundaries and the only six of the Indian innings.
Rahane fell two runs short of what could have been his fourth test hundred, playing aggressively and hitting Shakib for back-to-back boundaries before losing his off-stump to the next ball while attempting a pull shot.
(Reporting by Amlan Chakraborty in New Delhi; editing by Sudipto Ganguly)
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