Nathan Redmond had the first chance as he found space in the penalty area, but fired low and wide from a tight angle.
Nikola Zigic then fired wide at the other end before Grzegorz Sandomierski made a fine stop to deny Redmond from distance.
Chris Burke was next to try his luck from distance but both sides struggled to create openings in the final third for much of the first period.
The deadlock was broken just before the break as Ravel Morrison finished neatly from just inside the penalty area at a tight angle.
Blackburn responded positively after the break and equalised through Rhodes, who nodded home his 30th goal of the season from close range to level the scorers.
City searched for a late winner but were kept at bay by Sandomierski, as Rovers staved off any unlikely threat of relegation with a battling point.