Eastwood struck his 23rd goal of the season 90 seconds into the second half after Cobblers captain Chris Willmott had brought down Nicky Nicolau.
The Shrimpers will meet Lincoln City on Saturday in their second final of the season after losing in the LDV Vans Trophy to Wrexham.
Lawrie Dudfield went close to breaking the deadlock for the home side in the 24th minute but was denied by Lee Harper, after Eastwood had shaved the far post with a low shot.
Northampton were quick out of the traps and Charlie Hearn should have done better in the third minute but fired straight at Darryl Flahavan before having a late shot deflected wide.
Hearn almost took the game into extra time late on with a rocket which was deflected wide of the post.
And the visitors were denied in the final minute by a point-blank save from Flahavan to keep out Scott McGleish.
Southend winger Nicolau went close in the 12th minute after a blistering run upfield only to scuff his shot wide of the far post.