De is indeed a preposition here. This usage isn't unique to suivre, but can appear whenever you want to say you did something with your eyes:
Je l'ai vu de mes propres yeux (I saw it with my own two eyes)
Je l'ai confirmé de mes yeux (I confirmed it with my eyes)
Il m'a foudroyé des yeux (He glared at me, lit. He lightninged me with his eyes)
This isn't unique to the eyes either. There are a great many expressions in French that use de to mark the manner in which an action is done, the cause of the action of a reflexive verb or the agent of a passive verb. They are usually translated by with or by in English:
Suivre des yeux (Follow with the eyes)
Montrer du doigt (Show with a finger // by pointing a finger)
Frapper d'un poing (Strike with a fist)
Assomer de travail (overwork, lit. Stun with work)
S'étrangler de rage (Choke with anger)
S'éprendre de quelqu'un (Fall in love with someone)
Être ivre de victoire (Be drunk with victory)
Être aimé de tous (Be loved by all)
In Latin, those "de-noun phrases" would have been in the ablative case instead:
Oculis abeuntia vela prosequi - Ovid ("He chases departing sails with his eyes")
...sed digito gulam ei monstrabat - Visio Baronti (some Late Latin for a change, dd. 678 or 679 CE) (...but with his finger he pointed at [the other man's] throat)
Hic ita vixit, ut universis Atheniensibus merito esset carissimus - Cornelius Nepos' Three Lives, Atticus II 3 (Here he lived in such a manner that he was deservedly beloved by every Athenian)
Where oculis, digitō, and universis Atheniensibus are all in the ablative.
The ablative is named after a verb meaning "carry away from", but it had absorbed an earlier, separate case called the instrumental, used to mark the agent of a passive verb (by) or the object or mean used to accomplish an action (with).
The Latin ablative was replaced by dē in Vulgar Latin and its instrumental usages were ultimately inherited by de in Modern French.
In English (where the Instrumental had survived until Old English) these usages were taken over piecemeal by a variety of prepositions (of was used to mark passive agents in Middle English for example) until by and with won out.
De is concurrenced by avec and par for some of these instrumental uses, but it will nevertheless remain one of the main marker of means in the French language for the foreseeable future.