Bailey Gooda was back from his heel injury to replace Lewis Turner who has joined Farsley Celtic, and Theo Hudson came in for his first start in place of James Cadman, while Grantham included ex-Boro defender James Williamson.
In truth it was a scrappy and over-physical game, but Boro, with Kieran Glynn playing in a central midfield role created more scoring opportunities, and striker Nathan Cartman could have had a hat-trick, but had to settle for scoring the matchwinner to open his account for the season.
Three early corners for Boro gave hope to the travelling fans, and from the third, Nick Hutton had a low shot saved by the dive of keeper James Pollard. Grantham winger Manny Kianga shot straight at Ryan Whitley, then at the other end Hudson tried a theatrical scissors kick, but the ball flew wide of the upright.
Boro went ahead on 23 minutes when Glynn was clipped from behind in the area, and Ryan Watson sent the keeper the wrong way from the penalty spot, and five minutes later Watson flashed a free kick past the right-hand post.
Numerous high balls and wayward passes from both sides had the crowd frustrated, but on 40 minutes Glynn won possession and passed to Cartman, who beat a defender and fired a shot against the far post, then just before the interval a free kick into the Boro goalmouth was headed over the bar by Jonathan Wafula, when it looked easier to score.
Grantham began the second half on top and deservedly drew level on 53 minutes when Ash Jackson was adjudged to have fouled Harvey Grice, and the referee again pointed to the spot, with Michael Williams despatching his kick into the bottom right corner.
Gooda then deflected a shot from Rio Molyneux wide, but Boro then gained the upper hand and regained the lead in the 72nd minute with a well-worked goal.
A long throw from Jackson was headed into the six yard box by Gooda, and Cartman nipped in to flick the ball past keeper Pollard to register his first goal of the season.
Watson had a shot saved, then Grantham striker Wafuka blazed an effort wide, before Boro almost wrapped up the points late on, when Kieran Weledji laid off to Cartman, who curled a shot towards the top right corner, but keeper Pollard tipped the ball over the bar.
Gooda and Thornton were excellent in defence for Boro, while Glynn and Cartman linked up well in attack, Hudson impressed on his first start, and the two subs Weledji and Josh Barrett both made good contributions.
Not a great game, but three much needed points, and two big positives for manager Greening were hard-working striker Cartman getting on the scoresheet, and wing-back Barrett getting his first action of the campaign following his injury in pre-season.
BORO: Whitley, Gooda, Jackson, Heslop, Thornton, Cogill, Hudson (Weledji 74), Watson, Cartman, Glynn, Hutton (Barrett 61).
MAN OF THE MATCH: Bailey Gooda