A slow start from the young Rovers saw them go a goal behind to a strong City outfit after just five minutes, when a long range effort from Olivier Ntcham wasn't closed down and it crept home.
Despite that, Rovers defended really well in the first half, but couldn't retain possession well enough to make any meaningful impact on the City goal.
However they started the second half brightly and equalised on 51 minutes. Trialist Chike Kandi reacting quickly after Jack Doyle's free-kick was touched onto the post to fire home.
After that the game could have gone either way but it was City who grabbed the all-important second goal which would seal the victory.
Fine attacking play by Bersant Celina saw the winger cut in from the right before beating Callum Williams with an outstanding left-foot shot that went in off the far post.
"The boys played with a lot of commitment and honesty," said coach David Lowe.
"Again it will stand them in good stead for the remainder of this season and next year pitting their wits against the top team in our area at the moment.
"Unfortunately we couldn't create enough chances when we had good possession around their area and paid the price for not blocking their shots from distance.
"It was also good to see Devarn Green back after a lengthy lay-off."
ROVERS: Williams, Nyambe, Langford, Edgar, Lavelle, Hardcastle, Cham (sub Callaway 85 mins), Doyle, Kandi, Joel, Wall (sub Green 64 mins).
Subs not used: Fisher, Tomlinson, Wharton.