Hull took the lead in the 16th minute with Dan Batty bursting into the box and cutting the ball back for Eaves to score.
Rotherham were level five minutes later thanks to Michael Smith's powerful finish from Chiedozie Ogbene's centre.
The home side were reduced to 10 men in the 24th minute when Adam Thompson was given a straight red card for hacking down Keane Lewis-Potter.
Kyle Vassell then stunned Hull in the 43rd minute by firing in a thunderbolt from the edge of the box to give the hosts the edge.
The visitors' pressure paid off in the 66th minute with Eaves on hand to head in from Herbie Kane's cross.
Rotherham's resistance lasted until the second minute of added time, when Eaves was again on hand to punish them with a towering header and send the Tigers through.