But it looked like the Shrimpers would have to settle for a point for long periods at a snowy Roots Hall as Shrewsbury frustrated Steve Tilson's men.
The Blues were quick out of the blocks but it was the Shrews who had the best chance of the half after just six minutes.
Martin O'Connor's cross-field ball was centred by David Edwards to Kelvin Langmead, who saw his header parried by Darryl Flahavan. But Luke Rodgers wasted a golden opportunity by blasting the follow-up over the bar.
That proved to be the visitors only shot on target as Southend quickly responded with Lawrie Dudfield's header being held by Scott Howie, who then gathered a deflected Spencer Prior header.
Shrewsbury had a sustained period of pressure on the half hour mark but failed to capitalise before Eastwood replied by sending a rasping 20-yard shot inches over the crossbar on 39 minutes.
The hosts upped the pressure after the break but found Howie on fine form for the Shrews.
The custodian denied Mark Bentley, Mark Gower and Eastwood twice as it looked like Shrewsbury would stretch their unbeaten run to five games.
But as a snowstorm descended on Roots Hall it was the former Grays Athletic striker who got the breakthrough.
Substitute Duncan Jupp's cross was flicked on by Che Wilson and Wayne Gray to Eastwood, who nodded home from close range after 78 minutes.
The game threatened to descend into chaos minutes later when referee Joe Loss stopped proceedings to allow for an orange ball to be found as the blizzard worsened.
But officials were unable to find one so the match restarted again with the yellow match ball still being used.