The 22-year-old marksman scored in the 11th and 24th minute for the visitors while Luxembourg replied in the 46th minute though Lars Krogh Gerson's pin-point free-kick.
The match was their first since the departure of Craig Levein with interim manager Billy Stark, promoted from the Under-21s, guiding the team to a much-needed win.
Rhodes could have opened the scoring in the eighth minute following a mistake by defender Guy Blaise, but he shot wide with only keeper Jonathan Joubert to beat from the edge of the penalty area.
The Blackburn striker did find the back of the net two minutes later when Ante Bukvic failed to deal with Paul Dixon's cross, giving Rhodes the simple task of heading home from close range.
Luxembourg had chances. Firstly Matthew Gilks saved Mario Mutsch's back post header while Charles Leweck's shot after Christophe Berra's slip was poor and well off target.
Scotland doubled their lead in the 23rd minute when Andrew Shinnie was allowed the cross in the box. Joubert could only parry the effort which fell kindly for Rhodes to slot home.
Gilks was again called into action before half time. The Blackpool shot-stopper easily saved Gilles Bettmer's effort from distance before blocking Leweck's drive.
Luxembourg pulled a goal back two minutes after the interval. The home side made the most of a foul by substitute Liam Kelly for Gerson to curl the resulting free-kick beyond Gilks.
The teams made a host of changes in the second half and it was substitute Stafano Bensi who came close to equalising in the 71st minute. He connected with Mutsch's cross, but Gilks, albeit unconvincingly, scrambled the ball to safety.
For a second Rhodes thought he had secured his hat-trick in the 83rd minute when he guided a header into the net, but the offside flag was already raised.| Luxembourg | Team Statistics | Scotland |
| 1 | Goals | 2 |
| 0 | 1st Half Goals | 2 |
| 7 | Shots on Target | 3 |
| 4 | Shots off Target | 4 |
| 2 | Blocked Shots | 4 |
| 3 | Corners | 3 |
| 22 | Fouls | 18 |
| 1 | Offsides | 4 |
| 0 | Yellow Cards | 0 |
| 0 | Red Cards | 0 |
| 79.7 | Passing Success | 83.4 |
| 14 | Tackles | 8 |
| 85.7 | Tackles Success | 87.5 |
| 42.1 | Possession | 57.9 |
| 48.8 | Territorial Advantage | 51.2 |
| 379 | Total Passes | 518 |
| 17 | Total Crosses | 21 |
| 138 | Lost Balls | 148 |
| 54 | Recoveries | 61 |
| 42 | 1st Half Poss. | 58 |
| 42.7 | 2nd Half Poss. | 57.3 |
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