The Wales international was a key part of Burnley's promotion-winning side last season, scoring 20 Championship goals, but the striker sustained a serious knee injury in March that kept him out for nine months.
In only his second senior appearance since that injury, Vokes came on as a second-half substitute and turned home Michael Kightly's cross to ensure his side remain in contention to face Leicester City in the fourth round later this month.
In a relatively low-key affair, Nacer Chadli had earlier given Spurs the lead with a composed finish in the 56th minute, but the home side responded strongly and secured a replay through Vokes' strike.
A repeat at White Hart Lane was possibly not the desired result for either side, especially Spurs, who are still challenging in the League Cup and the UEFA Europa League.
Burnley, meanwhile, are now winless in five matches in all competitions, but Sean Dyche's side may take heart from another comeback after showing their battling qualities in recovering from losing positions against Manchester City and Newcastle United in their last two outings.
Despite a gruelling festive period, the home side made just two changes from the side that started their 3-3 draw at Newcastle last time out.
Captain Jason Shackell picked up a calf problem in that game and missed out, while David Jones was rested, with Daniel Lafferty and Kightly coming into the team.
By contrast, Spurs made seven alterations from their thrilling 5-3 defeat of Chelsea as Mauricio Pochettino looked to give some game time to his fringe players.
After a first half that failed to yield a shot on goal from either side, Pochettino introduced the in-form Harry Kane from the bench at the interval in place of Benjamin Stambouli.
Perhaps boosted by his arrival, Spurs threatened at the start of the second period as Federico Fazio brought a comfortable save out of Tom Heaton before Kane shot wide of the target.
Burnley responded as Danny Ings pulled a long ball out of the air and tested Michel Vorm with a drive from distance.
But Spurs went ahead 11 minutes into the second half as Chadli controlled Ben Davies' cross and fired in left-footed from close to the penalty spot.
With the home side reeling, Heaton had to be alert to keep out Kane's shot after Roberto Soldado had played his strike partner in.
Dean Marney fired off target as the game entered the final 20 minutes, before Vokes, who had replaced Ashley Barnes 12 minutes earlier, converted Kightly's cross with 17 minutes remaining.
In a bid to avoid a replay, both sides pushed forward in search of a winner but one was not forthcoming, meaning the pair will clash again in north London to decide who faces Leicester.
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