County, who rested four first-choice players from their starting line-up, had done well to contain Rovers for an hour and they upped their game when Weston came off the bench and were rewarded for their efforts.
The former Charlton winger instantl
y began to cause problems with his pace and trickery and when his chance came after 106 minutes following a neat exchange of passes with Jay Smith, he took it superbly with an angled drive.
Doncaster had been the more menacing team in the first half, Kevin Pilkington turning over a shot from Mark Wilson and a free-kick from Gareth Roberts clipped the top of the crossbar.
But County began to create openings after the break and after Roberts had denied youngster Ben Fairclough with a superb saving tackle, Mike Edwards squandered a great chance to clinch victory after 88 minutes.
Neil MacKenzie set up the County captain, but he fired over from the edge of the six-yard box when he only had Rovers' goalkeeper Neil Sullivan left to beat.
However, Weston made sure it was not a costly miss with his clinical strike early in the second half of extra-time.
Notts County: Pilkington, Beardsley, Edwards, Tann, Mayo, Fairclough (Hamshaw 69), MacKenzie, Smith, Butcher, Weir-Daley (Weston 66), Facey (Canham 84). Unused substitutes: Hoult, Johnson, Forrester, Nowland.
Doncaster Rovers: Sullivan, Chambers, Van Nieuwstadt, Hird, Roberts, Wilson, Wellens (Woods 77), Spicer (Coppinger 77), Guy, Heffernan, Hayter (Elliott 61). Unused substitutes: Smith, O'Connor, Mills, Lockwood.
Referee: M Oliver (Northumberland).
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