The former Ipswich striker made the telling contribution with his first real chance of the game midway through the second half with an assured low drive.
And his strike partner Rudy Gestede headed in a second with 10 minutes remaining to end a run of four successive draws and give them a first win at Ewood Park in over a month.
The Tractor Boys were always in the tight game and had Daryl Murphy got a header on target in the first half, the outcome might have been different.
But the result ends their run of two away games without conceding, and they are five points from sixth place.
Gary Bowyer opted for more firepower after the draw against Brighton and recalled Gestede for David Dunn in his only change.
Ipswich boss Mick McCarthy made two changes from the draw with Nottingham Forest. Paul Anderson missed out through injury but Anthony Wordsworth returned.
The visitors should have been in front after 26 minutes when Murphy beat Paul Robinson to the ball with a towering header to meet a Jonathan Williams cross but he thumped over the open goal.
Tom Cairney's first appearance since February last time out yielded a goal and an assist and the former Hull man went close here with a bending shot that Gerken gathered low to his right.
Gerken had to be at his best after 37 minutes to tip away Michael Keane's glancing header while at the other end, the visitors broke quickly and Williams lashed goalwards to force Robinson into a fine save.
Williams was the bright spot for an Ipswich side looking to win at Ewood in the league for the first time since 1991 and he nearly gave the Tractor Boys the lead with a stunning run on the break that saw him waltz past Evans but his low effort drifted wide.
Robinson was called into action after 58 minutes when Frank Nouble let fly from 30 yards and he produced an acrobatic save to tip away.
At the other end, Conway was set up by Gestede in the area but lashed his ferocious effort over.
But it did not matter as they went in front after 63 minutes when Ipswich old boy Rhodes latched on to Gestede's lovely flick before clinically firing into the bottom right corner from the edge of the area.
Gestede should have made it two with 15 minutes remaining when Rhodes hooked goalwards but his strike partner blazed over from point-blank range.
But he secured the win for Rovers minutes later by heading home Craig Conway's free-kick through a flurry of players for his sixth goal for the club.