Eastwood rolled home the spot-kick on 49 minutes after the impressive Nicky Nicolau was brought down in the box.
The first half saw few clear chances, although Southend's Mark Bentley did force a smart save from Lee Harper.
In a frenetic ending, Darryl Flahavan produced a brilliant point-blank stop to deny Northampton's Scott McGleish.
Northampton boss Colin Calderwood on Southend's penalty:
"It's a huge let down for everyone here - I saw the penalty incident in live action and questioned it but I'd have to have another look," said Calderwood.
"We'll gnash our teeth and have a long summer to think about it but you have to be sure to give decisions like that.
"We didn't do certain things well and that meant we were chasing the game."
Southend: Flahavan, Jupp, Nicolau, Barrett, Maher, Bentley, Pettefer, Dudfield (Gray 72), Prior, Wilson, Eastwood (Guttridge 90).
Subs Not Used: Griemink, Edwards, Gower.
Booked: Barrett, Bentley.
Goals: Eastwood 49 pen.
Northampton: Harper, Chambers, Willmott, Murray (Hunt 64), Kirk, Low, McGleish, Rowson (Richards 79), Hearn, Smith (Sabin 56), Togwell.
Subs Not Used: Bunn, Williamson.
Ref: B Knight (Kent).