No, they are not directly cognates. The -ate comes from Latin -atus, as Roux stated above, and the -atus is dividable in -a and -tu(s), as Nikolay stated. -tus is homonymic for participle *-to and verbal noun *-tu. But the verbal noun *-tu is not the same verbal noun as verbal noun in *-ti(s), whose dative *-tei gave us our -ть- and -ти-infinitives (-чь-infinitive is -к-ть or -г-ть). The -a- of Latin verbs and of Russian verbs is usually suffix in both (and even the exception is the same: dare/дать), but it is not always the same one (though there is verbalisator -ā in both languages). So neither part is directly cognate.
I also think it is worth mentioning that when we have loanwords for verbs in -ate, they usually end in -ировать (because of German and French interference), as in castrate - кастрировать, meditate - медитировать. However, their verbal noun is -ация for -ation: castration - кастрация, meditation - медитация.