A first-half strike from Exiles midfielder Ben Tozer and a second-half penalty from Morecambe captain Danny Rose ensured the points were shared.
Newport opened the scoring when Tozer tapped home after collecting a through ball from Frank Nouble.
But the visitors were a threat on the counter-attack, with Scot Bennett shown a yellow card for a cynical pull-back on Adam Campbell.
Padraig Amond came close to extending the home side’s lead on the stroke of half-time but his close-range header was cleared off the line.
The Shrimpers applied far more pressure in the second half and were awarded a penalty when Garry Thompson was brought down in the penalty box.
England manager Gareth Southgate has claimed that the country needs to boot racism out of its own football before criticising World Cup hosts Russia.
Newcastle have been linked with a shock move for former Sunderland goalkeeper Simon Mignolet during the summer.