The Collaborative International Dictionary of English v.0.48:

Convince \Con*vince"\, v. t. [imp. & p. p. Convinced; p. pr. &
   vb. n. Convincing.] [L. convincere, -victum, to refute,
   prove; con- + vincere to conquer. See Victor, and cf.
   Convict.]
   1. To overpower; to overcome; to subdue or master. [Obs.]
      [1913 Webster]

            His two chamberlains
            Will I with wine and wassail so convince
            That memory, the warder of the brain,
            Shall be a fume.                      --Shak.
      [1913 Webster]

   2. To overcome by argument; to force to yield assent to
      truth; to satisfy by proof.
      [1913 Webster]

            Such convincing proofs and assurances of it as might
            enable them to convince others.       --Atterbury.
      [1913 Webster]

   3. To confute; to prove the fallacy of. [Obs.]
      [1913 Webster]

            God never wrought miracle to convince atheism,
            because his ordinary works convince it. --Bacon.
      [1913 Webster]

   4. To prove guilty; to convict. [Obs.]
      [1913 Webster]

            Which of you convinceth me of sin?    --John viii.
                                                  46.
      [1913 Webster]

            Seek not to convince me of a crime
            Which I can ne'er repent, nor you can pardon.
                                                  --Dryden.

   Syn: To persuade; satisfy; convict.

   Usage: To Convince, persuade. To convince is an act of
          the understanding; to persuade, of the will or
          feelings. The one is effected by argument, the other
          by motives. There are cases, however, in which
          persuade may seem to be used in reference only to the
          assent of the understanding; as when we say, I am
          persuaded it is so; I can not persuade myself of the
          fact. But in such instances there is usually or always
          a degree of awakened feeling which has had its share
          in producing the assent of the understanding.
          [1913 Webster]

WordNet (r) 3.0 (2006):

convince
    v 1: make (someone) agree, understand, or realize the truth or
         validity of something; "He had finally convinced several
         customers of the advantages of his product" [syn:
         convert, win over, convince]

Moby Thesaurus II by Grady Ward, 1.0:

46 Moby Thesaurus words for "convince":
   argue into, assure, be convincing, bring around, bring home to,
   bring over, bring round, bring to reason, captivate,
   carry conviction, charm, con, convert, convict, draw, draw over,
   drive home to, evangelize, gain, gain over, get, hook, hook in,
   inspire belief, lead to believe, outtalk, persuade, prevail,
   prevail on, prevail upon, prevail with, prompt, proselyte,
   proselytize, satisfy, sell, sell one on, sway, talk into,
   talk over, wangle, wangle into, wean, wear down, win, win over