First-half strikes from Kane Ferdinand and Ryan Hall put Blues ahead before club legend Eastwood got his first goal since returning on loan from Coventry. Bilel Mohsni completed the rout as Southend recorded their biggest home win of the season.
Cheltenham suffered an early blow when referee Trevor Kettle had no hesitation in sending off Sido Jombati for fouling Michael Timlin after 19 minutes.
Nine minutes later the 10-man visitors went a goal behind when Hall's cross from the left was nodded on by Eastwood to Ferdinand at the back post and he fired into the bottom left corner from five yards.
Southend doubled their advantage in the 39th minute when Hall cut in off the right wing and unleashed a 30-yard rocket which went into the centre of Jack Butland's goal after the keeper misjudged the shot.
Eastwood made it 3-0 in the 51st minute when he latched onto an upfield Hall punt, beat lan Bennett and saw his deflected shot find the top-left corner. Butland was beaten for a fourth time in the 67th minute when he failed to deal with Mohsni's 25-yard effort which went straight down the middle.
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