The big English striker managed to pop up in the 68th minute to finish at the second attempt from close range after Barry-John Corr had pulled off a tremendous point-blank save to give the home side all three points.
Accies manager Billy Reid had Saturday's match against St Mirren in his thoughts as he fielded an unfamiliar side, handing a debut to Marc McJimpsey in centre midfield with Alex Neil reverting to a sweeper role in a back three.
Both sides struggled as a result to make any sort of impact in the first half with only a left-foot volley from Juanjo illustrating any sort of attacking endeavour from either side.
The second half did not serve up a more appetising platter although Accies striker Pat Keogh had a great chance from six yards but the ex-Celtic man skewed his shot wide just after half-time.
Stranraer gave the home side a fright when Leigh Hinds tried his luck from 20 yards. Accies keeper David McEwan spilled the shot but Ross McCabe was on hand to clear the danger.
Juanjo almost gave Accies the lead just before Wake pounced for the game's only goal, but after a mazy run, the on-loan Spaniard crashed the ball off the upright from 16 yards.
Stranraer pressed for an equaliser but Hamilton were quick to counter with the pacy Mark Gilhaney providing all sorts of problems for the away defence.
The Blues almost stole a point with only a minute remaining when Payne's corner was met by David Hamilton but the midfielder's header was acrobatically tipped over the bar by McEwan.
The away side ended the match with only 10 men as Hinds was sent off for initially fouling Gilhaney to earn a caution before verbally abusing the referee and his opponent to earn a second yellow.