Gilot converted a Jimmy Muitt cross from close range in the fourth minute and Pearce produced a sublime lob to double the lead before the break.
Braintree made Bognor sweat by pulling one back midway through the second half but the Rocks dug in to claim their first home league victory since August 15.
It will give them new hope that they can stay up - but more wins are needed, and fast, as they play catch-up with the group of teams above them near the bottom of National League South.
See a slideshow of Tommy McMillan's pictures from the match, above
The Rocks began a sunny afternoon at Nyewood Lane hoping to find that elusive home league win - something they last managed on August 15 in the second week of the season. They were boosted by the return to the starting line-up of striker Pearce, who has been out since November with a troublesome groin problem. But they were without midfielder Dan Beck, who picked up a groin injury at Chippenham a week ago. Braintree arrived at the Nye Camp in the thick of the play-off race, sitting eighth on 46 points but only four points off third-placed Havant.
The Rocks let one in only 21 seconds into the game at Braintree in November - they at least avoided such an early setback this time as the Essex side soon lost possession after kicking off.
Bognor have found success at home so hard to come by all season but they went ahead in the fourth minute thanks to a close-range effort b Gilot, his first goal for the club. Muitt made it by getting clear on the left and taking his time before crossing low across the six-yard box and Gilot poked it in at the second attempt despite a defender's attentions.
The Rocks looked the better side after that and put the visitors' goal under threat a couple more times before Braintree managed their first shot at goal - and that was a Reece Grant effort that drifted well wide and didn't trouble Dan Lincoln.A long ball forward to Billy Crook had the Rocks worried but he couldn't get a decent connection on his shot and Lincoln gathered.
Bognor played themselves into trouble on 21 minutes and Braintree debutant Connor McKendry had sight of goal but Lincoln got down well to save. McKendry gave Bognor a scare on the half-hour when he ran through and squeezed a shot past Lincoln, but Keaton Wood was alert to it and raced back to dribble the ball off the line then clear.
Braintree's Joshua Hill was booked for a cynical foul on Doug Tuck in the centre of the park. Braintree were employing some rough tactics and at one stage skipper Marc Okoye was spoken to by ref David Spain, who clearly wanted him to control his team and stop the persistent fouling.
On 40 minutes the home fans erupted again as Pearce marked his return to the starting line-up with a lovely goal - lobbing the advancing keeper from the edge of the area after taking and controlling a pass into him by Gilot. No-one could quite believe it - this was the first time the Rocks had scored two at home since September 23. HT 2-0
Pearce raced clear on to a Muitt pass a minute or so into the second half but an offside flag shot up. Five minutes into the half Tommy Block sent Muitt racing away down the right and after beating his defender, he fired a fierce shot goalwards that Nathan McDonald did well to palm away. Then Harvey Whyte played in Gilot but his shot, though on target, was half-blocked.
Muitt charged down a defender's clearance and got forward to send in a nice cross from the right just before the hour mark but the visitors dealt with it. The first substitution of the afternoon came on 61 minutes as Roman Michael-Percil replaced Grant, and Ben Swallow came on for Pearce.
Braintree got back in it on 64 minutes from nowhere as an attack down the centre ended with Crook curling a nice low shot beyond Lincoln's dive and into the corner from 18 yards.
Whyte was booked for stopping Braintree taking a quick throw-in. Braintree threatened an equaliser when Michael-Percil danced past three defenders on a run across the 18-yard line but shot straight at Lincoln. Swallow ambitiously tried a shot from 25 yards after cutting in from the right but it was all along the floor to the keeper.
Jonathan Muleba whistled a drive not too far wide from outside the area with 15 minutes left. Braintree brought on Matt Baxter for Luke Allen with12 minutes left, before Muitt's afternoon of hard work ended as he was replaced by Ibra Sekajja. Braintree forward Marcel Barrington was booked for a clash with Manny Adebowale as the Essex men tried to make progress down the left. Soon skipper Okoye was booked too for a nasty foul on Whyte.
A quick Rocks counter-attack almost brought a third. Wood's ball to Swallow saw him fire in a fine cross from the right and when it was cleared, Tuck teed up Gilot for a volley that went just wide. Braintree had very little threat of their own in the final few minutes and the Rocks comfortably held on through four minutes of injury time for three valuable points.
Next they go to Hampton and Richmond in a week's time - and they do so with new belief.
Rocks: Lincoln, Whyte, Campbell, Tuck, El-Abd, Adebowale, Gilot, Block, Muitt, Pearce, Wood. Subs - Swallow, Sekajja, Charman, Scutt, Nelson.Braintree: McDonald, Muleba, Webber, Hill, Okoye, Wyatt, Allen, Crook, Borrington, Grant, McKendry. Subs - Frimpong, Bettache, Baxter, Michael-Persil, Curtis.Ref: David Spain (Westfield)Att - 558