Disclaimer - From a Wesleyan-Arminian soteriological perspective:
Although they were both pronounced 'very good' (I will avoid using the apparantly contentious word 'perfect' - although 'good' alone is potentially just as contentious - until addressing it's use after the close of my argument), Adam and Eve's choice to sin was a tragic* inevitability of the interplay between their being granted free will, and God's Sovereign determination to outwork his plan of redemption.
Firstly regarding God's plan, scripture tells us that Christ is the "Lamb slain from the foundation of the world." (Rev 13:8) - i.e. it was always going to be this way. God acheives his greatest Glory through the plan of redemption by fully demonstrating the two great and quintessential aspects of his nature: His Holiness (Is 6:1-5) and His love (1 Jn 4:7-16). The plan of redemption necessarily involves redeeming - via an extraordinary act of Love - a rebellious people from the consequence of the sin that has enstranged them from the most Holy God. Even though there was a necessity to man's fall, the way it was accomplished did not undermine man's free agency as: God presented an opportunity for obedience or disobedience inherant in the tree of the knowledge of good and evil, but he neither compelled one nor the other (in fact that disobedience was chosen despite God's command and warning is sufficient testament that man was a free agent). He (God) further allowed another free agent to act as tempter and start the chain of unfortunate events (scripture teaches us the Serpent was animated by Satan to do the particular work - cf. Rev 20:2). That this was foreseen and permitted as working with His plan (as opposed to Sovereignly and irresistably decreeing it) is deduced from his disavowal of tempting anyone (James 1:13)
In terms of the outworking of this plan with respect to the free will of Adam and Eve, it manifested in a different manner (or more accurately, sequence) for each of them:
Firstly, Eve in being deceived, was not (initially) fully culpable. This is born out in the judgment applied to her: she was disciplined for her error, but the death sentence was not pronounced against her. (cf. Genesis 3:16) Adam, however, sinned wilfully and had the death sentence passed against him (it's effect was immediate in spiritual terms - being cut off from God's presence - and eventual in physical terms - he would, after a life of toil, return to the dust). The Romans passage emphasizes that the sin of Adam (not Eve, nor even Adam and Eve) was the cause of the fall. It was he that was barred from the Garden and the tree of life not Eve. But just as Adam wilfully followed Eve's error into sin, Eve wilfully chose to leave God's presence and follow Adam into sin and death after the judgments were pronounced.
The question remains "why did Adam choose to sin?", He himself answers - "because of the woman" (v12). Modern eyes may see this as mere blame-shifting, but in the absence of other evidence, (and by adopting a hermeneutic compatible with 2 Tim 3:16) we can assume that this is actually the true reason. Adam chose to follow Eve rather than God's command out of a considered choice motivated by desire. He could not have understood the full implications of 'death' and would perhaps have been unsure what would happen to Eve after she had been deceived and eaten the fruit, but he was fully resolved not to be separated from her, even if it meant the loss of everything else - he was the bible's first idolater (placing as an object of supreme affection, something other than God). This decision was possible for a 'very good' creature with free will to make - he did not have perfect knowledge to see the full implications of his decision, but in his eyes, it was a reasonable decision made with good motives - though he was (self-)deceived in thinking there was no better option, as the glorious plan of redemption was yet hidden to him. Likewise, though in a different order, Eve's decision to leave the garden with Adam and follow him into sin was motivated by her desire for her husband opposed to an implied permission to remain with God. A logical conclusion from her preceding judgment, subsequent action in accompanying Adam and eventual physical death, is that Eve's 'sin that leads to death' (cf. 1 Jn 5:16-17) wasn't in being deceived or even in acting from that deception, but it was like Adam, an act of idolatory in choosing her husband over God.
Summary: The fall ensued even though the actors were 'very good' (sinless, not victims of total depravity and in possession of free moral agency) because they freely chose to play the parts written for them by the Sovereign Lord's intricately crafted tragedian* plot - the necessary counterpoint to his subsequent stunning redemptive revelation.
Regarding the use of the word perfect: as has been pointed out, there is often in it's use - in biblical terminology at least - the sense of completeness or full maturity, and in this sense, it is not legitimately applicable to Adam and Eve, as they were incomplete and lacking in full maturity in the area of knowledge at the very least. That it is not universally used in this sense in scripture is evident from the counter example of Lev 22:21 - many versions do not scruple to use the word here where it has the mere sense of flawless or unblemished (other versions translate the original thus) - which are entirely legitimate synonyms for perfection according to normal English usage. That the corresponding original should not be so translated to perfect in this instance is more a matter of theological preference than an artifact of language. Conclusion: it is not actually wrong to use the word in application to prelapsarian Adam and Eve in a certain sense, but in order to avoid confusion, the term should either be further explained or just plain avoided.
*Tragic more in the sense of Romeo and Juliet, and beyond the sense used when referring to The Scottish play .