Matt Godden gave Stevenage, who did the double over the Shrimps last season, the lead in the 37th minute after Stevenage took full advantage of some shocking Morecambe defending.
Dale Gorman was given time and space down the right to pick his spot and cross perfectly for Godden to head past Barry Roche from six yards out.
Ben Kennedy then added a second just before the hour with a special strike. The midfielder tried his luck from 30 yards with a left-footed curler that gave home keeper Roche no chance as it dipped under the crossbar and ended up in the back of the net.
Morecambe were given a late chance to get something from the game when they were awarded a penalty with three minutes to go. Liam Wakefield's right-wing cross hit the top of Charlie Lee's arm and the referee pointed to the spot but Paul Mullin's effort hit the left post and the chance went begging.
It was the second time the Shrimps hit the woodwork as Lee Molyneux's 25th-minute shot crashed against the crossbar after a mistake from Jones.
The same player saw an effort cleared off the line moments later by Fraser Franks but after Stevenage opened the scoring the visitors took control.
The home side forced a save from Jones in injury time as Kevin Ellison's shot looked set for the top-left corner but it was too little too late from Jim Bentley's side.