Just after the interval Eastwood cut in from the left before unleashing a shot that crashed against the Luton bar.
In stoppage time Eastwood looked certain to score after he rounded keeper Dean Kiely, but he struck the base of the post with his shot.
Southend keeper Darryl Flahavan made a series of saves to keep Luton at bay.
"It's frustrating when you don't win your home games.
"It is difficult to play the bottom side - they're capable of winning games and it could have turned out that way.
"I don't think confidence was lacking, but that was as poor as we've played this season. We gave the ball away and it was hard to get near their goal."
"It's a good point. We played some good stuff - on balance it's a fair result.
"At the death I thought we'd won.
"When you're at the bottom you don't get that luck, but two unbeaten games gives us a platorm to build on with so many games coming up."
Luton: Kiely, Foley, Edwards, Robinson, Heikkinen, Morgan (Feeney 65), Emanuel, Brkovic, Barnett, Bell, Vine (Coyne 81).
Subs Not Used: Beresford, Langley, Boyd.
Southend: Flahavan, Hunt, Prior, Sodje, Gower, Maher, Campbell-Ryce (Hooper 80), Hammell, McCormack, Eastwood, Bradbury (Harrold 80).
Subs Not Used: Welch, Francis, Clarke.
Att: 7,468.
Ref: N Miller (Co Durham).