They now need just two points from four matches to be sure of finishing at least fifth and so making the play-offs in National League South in their first season back at this level.
City took the lead after 22 minutes when Matty Jay followed up to score after Matt Wright's shot had only been parried by the Concord keeper.
And they doubled their advantage six minutes before half time when Shane White scored from the spot after the excellent Wright was tripped by Dowie on the right edge of the box.
But the Essex club hit back on the hour when Matt Greenhalgh broke through on the left and slotted the ball past the advancing Martin Rice.
But City held firm for a priceless win thanks in part to a superb last gasp save by Martin Rice, as they got back to winning ways after suffering two successive home defeats.
They will be hoping to wrap up a play off spot with victory at already relegated Basingstoke on Saturday when skipper Arran Pugh will be available after completing his two match ban.
On Tuesday they are at home to Whitehawk and then next Thursday travel to Havant & Waterlooville before completing their 42-match league programme at home to Wealdstone on Saturday April 30.
Tully said: "It was a big win and the changes had the desired effect. All those who came in did a great job.
"We could have been more ahead at the break and then gave them a rather soft goal. But we dug in when we had to and Ricey made a couple of crucial late saves when he was called into action.
"We are in fourth and that's where we want to finish."
To be sure of fourth City need five more points.
CITY: Rice; Mitchell (Cooke 69), Ash, Richards, White; Dawson, Adlesbury, Brett, Jay, Afful (Knowles 75), Wright (Vassell 69). Subs (not used): Green, McHale.
Goals: Jay (22) 0-1, White pen (39) 0-2, Greenhalgh (60) 1-2
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