All looked well at the interval with goals from Tom Berwick, his first for Stourbridge, and Jordan Fitzpatrick giving the home side a comfortable lead.
However, Evesham came out fighting after the break and reduced the arrears through Ellioot Kennedy, before Lance Smith levelled fifteen minutes from time.
There was an explosive start to the extra thirty minutes when Dean Coleman brought down an attacker and Jared Wilson netted from the penalty spot, and then Sean Geddes saw his spot kick saved at the other end.
Both teams had chances as the game opened up late on but there were no further goals and the Robins progressed to the Final.
The attendance at the War Memorial was 231. A full report will appear here in due course.
Meanwhile, results last night from the Calor League Premier Division...
Arlesey Town 2-3 St Albans City
Bedford Town 1-5 Poole Town
Chesham United 8-0 Banbury United
Hemel Hempstead Town 3-0 Biggleswade Town
Weymouth 3-2 Bideford
Stourbridge return to action on Saturday, when Bashley FC visit Amblecote for a League fixture. Kick off is at 3.00 pm.
Before then, the Youth Team are scheduled to entertain Ellesmere Rangers at home tonight (Wednesday 5th February) in the Staffs County Youth Cup. Kick off is at 7.45 pm. Supporters are advised to check before setting out due to yet another adverse weather forecast!