Spencer scored twice in the second half after seeing a first-half drive deflected in by a visiting defender.
The Silkmen still harboured faint hopes of automatic promotion before this match but they were left needing a result in their final game next week to make absolutely sure of a place in the play-offs.
Cheltenham, meanwhile, have had an inconsistent season but they showed what they are capable of with this powerful display.
The home side took the lead after 33 minutes when Grant McCann picked out Spencer with a fine pass from midfield. Spencer raced into the box from the right-hand side and his cross-cum-shot was deflected past his own goalkeeper by defender David Morley.
It had been a fairly even contest until then but in the closing minutes of the first half, Cheltenham had several chances to increase their lead.
Kayode Odejayi saw a firm on-target shot held by goalkeeper Steve Wilson, then Spencer headed a fraction wide of goal after meeting a McCann corner with an unmarked header.
Odejayi had an even better chance five minutes before the break when Brian Wilson found him in space with a cross from the right. However, the big striker was unable to guide his volley onto the target and it drifted wide.
Macclesfield struggled to create clear-cut openings in the first period. Their best chance was a volley by Neil Mackenzie that cleared the crossbar by some distance.
If Cheltenham had enjoyed a better time of it in the minutes leading up to half-time, they took control of the contest during a second half in which Macclesfield never really came out of their shell.
Visiting striker Jon Parkin forced a good save from goalkeeper Shane Higgs and Matthew Tipton hit the bar with an opportunist effort, although most of the second-half traffic was towards the Macclesfield goal.
Brian Wilson, JJ Melligan and David Bird all went close with chances before the second goal arrived on 64 minutes.
A mammoth goal kick from Higgs bounced once on its way towards the goal at the opposite end. Spencer jumped with the goalkeeper and reached the ball fractionally ahead of him, the big striker twisting to latch onto the loose ball and sending an overhead kick into the empty net.
Spencer completed the scoring 12 minutes from time when a McCann corner resulted in a goalmouth scramble and the striker pounced from close range to register his ninth goal of the season.