The 30-year-old, who played for Kilmarnock from 2004 to 2007 before moving to Chesterfield, MK Dons and Oxford United has overcome a knee injury to join the Roofers as a player/coach.
Leven already has a UEFA B Licence and will work towards his A Licence with Jarrow manager Richie McLoughlin and his assistant Ian Davison. McLoughlin said: "Peter's signing is a huge boost for Jarrow. Whenever I saw him play in the Football League I was impressed."
Stranraer failed to bring winger Willie Gibson to Stair Park when the 30-year-old opting instead to join Evo-Stik League Northern Premier side Workington Town.
Gibson played for Queen of the South, Kilmarnock, Dunfermline Athletic, Crawley Town, St Johnstone and Falkirk before heading for Celtic Nation in Cumbria. Aitken said: "I spoke to his agent on Monday; however he signed for Workington the next day, so that is that."