Harwich came close to scoring after only 6 minutes when Jenson Goode's pass split the Rovers defence and returning captain Alfie Canning's low drive was diverted for a corner but could just as well ended up in the net.
Twelve minutes later the Shrimpers came close again when Jake Gordon's angled free kick was only a few degrees off target.
Gordon featured again with a well-placed corner kick from which Lewis Wood's powerful header was cleared off the goal-line.
Twenty-eight minutes had gone when Rovers threatened to score for the first time but full-back Jeremiah Ajisafe made a last-ditch goal-line block.
A well-worked move involving Harwich's two Alfies, Eldridge and Canning, played in Joe Wood, which proved inches too far for the striker to reach and so the first half ended goalless.
Shrimpers central defender Charlie Canham had aggravated an old injury and so Tyler Sullivan took his place in the second half.
Seven minutes after the restart, Harwich took the lead when Tiwa Odufuye and Canning combined to find Joe Wood and in a crowded goalmouth, he did well to lay the ball off to Louie Bloom (pictured) who skilfully worked himself into a shooting position before firing low into the far corner of the net.
For Harwich, Joe Wood had a decent scoring opportunity but shot past the post, whilst at the other end, when Rovers manoeuvred the ball past Shrimpers goalkeeper Will Healy, the dominant, calm and collected Lewis Wood was well positioned to clear the danger.
Next, Healy pulled off an excellent save to turn a close-range shot away to safety.
Disaster struck Harwich in the 67th minute when, from a defensive pass across the goalmouth, the alert Jack Mitchell snapped up the gift to equalise for the Rovers.
The excitement was intense now as both teams sought a winning goal.
Influential midfielder Gordon, still seeking full fitness, had been replaced by young Shrimper Josh Hawkins, who was proving to be a very capable substitute with his energy and astute passing.
Bloom was a dynamo for Harwich, as was Eldridge, and they both came close in the quest for a winner.
Rovers are a team who suffered pre-season departures but are now knitting together a good team and they had a late chance from a free kick on the edge of the penalty area that went just over the crossbar.
Overall it was a really good showing from Harwich, in which they could have gleaned all 3 points, playing attractive, attacking football and also displaying fighting spirit when the chips were down.
Harwich team: Will Healy, Jeremiah Ajisafe, Charlie Canham, Lewis Wood, Tiwa Odufuye, Jenson Goode, Jake Gordon, Louie Bloom, Alfie Eldridge, Alfie Canning (captain) and Joe Wood. Substutues: Tyler Sullivan, Josh Hawkins, Charlie Carr, Munashe Chirimuuta and David Ejakita (all used).