Dhoni, 43, had been Chennai's skipper for 14 seasons since the start of the IPL in 2008, barring the period between 2016-2017 when the team was suspended. But the veteran wicketkeeper handed over the captaincy to Gaikwad last year.
Gaikwad suffered the injury during Chennai's game against Rajasthan Royals when he was struck on the elbow while batting but went on to score 63 runs. He also played the next two games against Delhi and Punjab.
The five-time IPL champions have had a poor start to the season, sitting ninth out of 10 teams after losing four of their first five matches.
Chennai's next game is at home against Kolkata Knight Riders on Friday.