The altercation was missed by the match officials but captured on video, with the FA saying the standard three-match suspension was an insufficient punishment.
"Connolly accepted the charge but denied the standard punishment was clearly insufficient," the FA said on its website on Wednesday following a hearing.
"However, the Independent Regulatory Commission found in favour of The FA and ordered that Connolly be suspended for five matches, commencing with immediate effect."
Brentford won the match 1-0 and Connolly can appeal against the decision.
(Reporting by Josh Reich; Editing by Ken Ferris)
