The Reds overwhelmed the visitors for the majority of the game but had to settle for a point after conceding a sloppy goal and squandering gilt-edged opportunities.
It took Gravesend just 92 seconds to take the lead thanks to new signing Craig McAllister.
The striker, who had joined from Stevenage a week earlier, out-muscled Reds captain Ian Simpemba on the edge of the box and beat Phil Smith with a delightful chip.
Crawley responded by dominating possession for the rest of the first half and they got back on level terms just five minutes later.
Mo Harkin did well to curl a cross from the right flank over to Neil Jenkins, who squeezed home a header at the back post from a tight angle.
Crawley should have gone ahead in the 15th minute when Danny Davidson somehow volleyed wide from five yards with only keeper James Pullen to beat.
Dan Marney then missed a similar chance moments later after he had done well to beat Paul McCarthy to the ball.
Marney was left holding his head again two minutes before the break after he had earned a penalty when his header struck McCarthy's arm.
Marney won a dispute with Harkin over who should take the resulting spot-kick but the former Brighton star could only strike the outside of the post with his tame effort.
Crawley had the better of the second period, but it was Gravesend who had the best chance to take all the points.
Manny Omoyinmi raced onto an under-hit back pass by Joe Dolan to go clear on goal but as he tried to lift the ball over the onrushing Smith, the Reds keeper pawed the effort to safety.