As Joonas said in his answer, the anti- found in anticipāre and anticipātiō seems to be merely a variant of the Latin adverb ante/prefix ante-, not a Greek prefix.
There seems to have been at least one more word that often had this variant of the prefix ante-: antistes, which Lewis and Short say is from the verb antisto, a variant of ante-sto. Apparently, scribes did fairly often replace the spelling antistes with antestes, but the spelling with "i" seems to be considered more standard or more common.
According to "Aspects of the phonology and morphology of Classical Latin" by András Cser,
This prefix appears as ante- or anti- without any discernable regularity:
(74) Words prefixed with ante-/anti
antesignani 'troops in front'
anteferre 'carry in front'
antistes 'high priest'
[...] It would be interesting to see how the final vowel of anti-/ante- behaves in the environment of lowering before [r] (see 18.104.22.168), but this prefix does not combine with stems beginning with [r] at all.
Using the PHI Latin Texts search, I was able to find an isolated example of antilucanam used in place of antelucanam in L. Iunius Moderatus Columella, De Re Rustica 22.214.171.124. This is mentioned in "Emendationes in Collumellam" by Vilelmus Lundström (in Eranos, 1907):
XI, 2, 12 antilucanam hunc locum editurus scribendum curaui cum S A. Vix enim crediderim librarios ante- in anti-, quod iis alienum esset, mutauisse; contra anti- in ante- ut sponte mutarent, consuetudo facile persuadebat (cfr I, 10 antestites S m. pr., quod librarius exemplari suo diligentius inspecto ipse correxit). Quam ob rem si quis infra 2. 55 antilucanis (ubi antelucanis S A R) restituere uelit, equidem neque alteri dissuadere neque tamen ipse facere ausim. Nam inter -ĕ- et -ĭ- uacillare auctorem ipsum credo (cfr beneuolentia et beniuolentia sat multis locis).
Unfortunately, I can't read Latin well enough to give a translation of this text, but it seems to me that Lundström says here that it was more common for scribes to change anti- into ante- than vice versa, so he doesn't think that we can say for sure that anti- in antilucanis is just a scribal error (although Lundström also doesn't seem to feel that he can completely rule out that possibility).
Here are my guesses about possible factors that contributed to the change of ante- to anti- in anticipāre and anticipātiō:
e and i are phonetically similar
Of course, e and i in Classical Latin were fairly similar phonetically, both being front non-low vowels. This may have made it easier in general for them to interchange. Another odd case of /i/ occurring where /e/ might be expected is "decim" in words related to decem.
In anticipāre, the following syllable also has i, so it seems possible to me that regressive assimilation might have played some role in the development of ante to anti- here. But it's very unclear to me.
i is common in internal open syllables in Latin
In an open syllable that is not the first or last syllable of a word, a short vowel not followed by /r/ or a consonant cluster containing /r/ will often be /i/, because of processes of vowel reduction that occurred in older stages of Latin. In the time of Classical Latin, this reduction of short vowels in word-internal syllables to /i/ did not apply as an automatic, exceptionless rule. But the existence of this type of weakening may be relevant to the development of the anti- form used in anticipāre.
We see this kind of vowel reduction for example in the "cip" part of anticipātiō, and in the second part of a number of prefixed verbs such as assideō < ad- + sedeō. We also see "short i" as a vowel coming after the first element of a compound in words like artifex, from ars and facio.
The use of "i" as a generic "linking vowel" in Latin compound words (e.g. "altispex" and "extispex") may have provided some analogical basis for changing ante- to anti- in anticipāre (and, apparently, antistes, where the vowel isn't even in an open syllable).
A possibly similar variant is mali- in place of male- in the word maleficus; Lewis and Short say that malificus occurs as a variant spelling used in manuscripts.
Some words that are morphologically related to ante have i
Etymologically, the e in ante actually comes from i, and there are still remnants of this in some words: de Vaan says
original i is still visible in antiae, in the inflexion of antēs, and in antīquus.
De Vaan also mentions the compounding form antid-, which he says is "probably on the analogy of postid".
I don't know if this is at all relevant to the use of "i" in anticipāre, but it seems vaguely possible.