The Dons looked to have given themselves the chance of atoning for last year's semi-final defeat to Queen of the South when Sone Aluko capitalised on some sustained second-half pressure.
But, with nine minutes left at an electric East End Park, Phinn bundled the ball home from close range.
Aberdeen had looked in command after a slow start, but Jimmy Calderwood will be angry with his men's failure to stop Phinn following up after Jamie Langfield had parried Jamie Mole's long-range effort.
Calderwood handed injury-jinxed Jamie Smith his first start since August as he made four changes following Tuesday's 1-1 Clydesdale Bank Premier League draw at St Mirren.
Tommy Wright was rewarded for his injury-time equaliser with a rare start in place of the injured Lee Miller, and Derek Young and Stuart Duff came in.
Dunfermline harried Aberdeen well in an even but scrappy opening spell and the home side created the first chance in the 16th minute.
The Dons left Phinn unmarked on the edge of the area as Stephen Glass took an outswinging corner, but the midfielder volleyed high over as he ran in.
Dunfermline had a better chance 10 minutes later when Mole's left-wing cross found Graham Bayne 10 yards out but he directed his header just wide with Langfield beaten.
Aberdeen were struggling to create anything from open play and Calderwood switched Smith from a central attacking role to right midfield, with Young moving inside to support Wright.
The visitors soon carved out their first real chance as the Pars for once failed to deal with a high ball to allow Aluko a sight of goal 12 yards out.
However, the ball would not come down quickly enough for the winger and Scott Wilson made an impressive block, although Aluko appeared to have his shirt tugged in the process of shooting.
Aluko was becoming more influential and Glass was booked for bringing him down as he ran at the heart of the home defence.
Charlie Mulgrew almost punished Dunfermline further as he curled the 30-yard free-kick on target but Paul Gallacher leapt to push it over.
Aberdeen piled on the pressure at the start of the second half. Greg Shields blocked Aluko's low strike after some penalty-box trickery from the Englishman, and Gallacher held Wright's close-range header after great wing play from Smith.
Only a break for treatment to Bayne's head wound relieved the pressure but it resumed once the striker had been replaced by Andy Kirk.
Dunfermline were sitting back and Aluko punished them in the 61st minute. The winger took Young's pass in his stride and fired the ball into the far corner from 20 yards.
The home side almost produced a swift response but a superb save from Langfield maintained Aberdeen's lead. Phinn once again ran on to Glass' corner and powered a header towards goal but the goalkeeper dived to his left to push it wide.
The hosts continued to fight back and Langfield saved long-range efforts from Kirk and Alex Burke.
Richard Foster came on for his first appearance since a bust-up with Calderwood three weeks ago as Aberdeen looked to have the game sewn up, but the 5,500 travelling fans were silenced in the 81st minute.
Langfield could not hold Mole's 25-yard drive and Phinn scrambled the ball over the line after Kirk had beaten the Aberdeen keeper to the rebound.
The Dons forced the Pars back but Wilson blocked from Mair to ensure both teams will meet again at Pittodrie on March 17.