Shayne Lavery and Karamoko Dembele were also on the scoresheet for Neil Critchley’s side at Bloomfield Road.
Despite leading 3-0 after 40 minutes, a Will Goodwin brace made the closing stages tense for the Seasiders, as they needed to stay switched on to claim the three points.
Both sides had early opportunities to open the scoring.
Goodwin forced an excellent stop from Dan Grimshaw at one end, while at the other Dembele smashed the ball against crossbar.
The first goal of the evening came in the 17th minute, with Lavery putting the Seasiders in front.
A cross by Dominic Thompson to the back post was met by Rhodes, who headed the ball back across the face of goal- where the Northern Ireland striker was waiting to finish.
Blackpool had a chance to double their lead shortly after, with Callum Connolly placing a header wide of the target from close-range.
Just after the half hour mark, Critchley’s side did have their second, as Rhodes claimed his goal.
Albie Morgan attempted a shot from distance with a free kick, but the effort was deflected.
Rhodes was alert to the situation, and reacted quickly to direct a header towards goal, which ultimately found its way into the back of the net due to the keeper being caught off guard.
It wasn’t long until Dembele added his name to the scoresheet as well, with the former Celtic youngster producing a curling effort from the edge of the box.
Ahead of half time, Goodwin pulled one back for the visitors, as Blackpool’s calls for offside were waved away by the officials.
Following the restart, Blackpool came close to adding another, with CJ Hamilton firing an effort straight at Luke Southwood after cutting in from the right wing.
Cheltenham did grow into the second half, and pulled another goal back through Goodwin heading into the final five minutes.
Despite the visitors’ late resurgence, the Seasiders were able to hold onto the three points.
Here’s how Critchley’s side performed: