The young striker opened the scoring on 44 minutes when he latched on to a long ball from Tom Walsh before finishing with aplomb.
And on 62 minutes he again showed his composure to double his side's advantage when a corner was knocked down in the box.
But despite creating a lot of chances throughout the game Rangers were pegged back when Thistle scored on 66 minutes meaning the Light Blues had to work hard to hold on to their lead in the closing stages.
Billy Kirkwood was pleased to see his team start this season's Glasgow Cup campaign with a 2-1 victory and praised McKinnon for his matchwinning display.
But the coach also believes his players must learn to be more ruthless when they get their noses in front.
He said: “In parts in was a good performance.
“The criticism we could level at the lads is that for the amount of possession we had we didn’t turn that superiority into goals.
“We had opportunities but to be fair to Partick they dug in and because we didn’t take our chances they were always in the game.
“At 2-0 we thought we had a bit of breathing space but again we squandered opportunities and gave away a silly goal.
“For the last 15 minutes it was a nervy but overall I thought we deserved to win and I was delighted for Andy.
“He’s still in our under-16 side but we gave him a chance last night and he took his two goals brilliantly.
“He is on our day release project that Craig Mulholland has set up so he is still at school but trains on a Thursday with us and also four or five nights a week.
“He’s been playing under-16 games so we though he would give him an opportunity in the Glasgow Cup and he did very well.”
The Glasgow Cup is played on a league basis with Rangers taking on Celtic, Partick Thistle, Clyde and Queen's Park home and away during the 2012/13 season.
The final will be contested by the top two teams in the league table at the end of all 40 ties.
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