Grant’s header gave Blues the lead at half-time and a flurry of Hannah Hampton saves helped the side to a point.
England international Lucy Staniforth captained the side, and Rachel Williams came back into the team in place of Connie Scofield.
Academy prospect Lily Simkin retained her starting place, following her first senior start against Chelsea last time out.
The game almost got off on the wrong foot for Blues after two minutes. Rebecca Holloway misplaced a back pass to Hampton, but Rosella Ayane clipped the post with a left footed effort.
Blues started to grow into the game as the half progressed. Staniforth played a neat one-two with Chloe Arthur, with the former having her low drive saved by Becky Spencer.
Marta Tejedor’s side took the lead after 20 minutes. Staniforth played a delightful long ball that matched the run of Grant, who headed beyond the Spurs keeper.
Staniforth fancied a goal of her own minutes later, but her attempted lob floated over the bar. Blues were the ascendancy and looking to double their advantage after a shaky start.
Holloway came close to adding a second for Blues. A neat free-kick routine ended with the centre-backs effort being tipped over.
Blues ended the half well with Claudia Walker forcing a fantastic save from Spencer from the edge of the area.
After a slow second half, Spurs grabbed an equaliser with 15 minutes left. Gemma Davison floated in a free kick which was met by Kit Graham who headed past Hampton.
Blues: Hampton, Simkin, Scott, Holloway, Jordan, Staniforth, Whipp, Arthur, Walker, Williams (Scofield 74’), Grant.
Unused subs: Brooks, Gregory, Powell, Jones.
Tottenham Hotspur: Spencer, Percival, Godfrey, Schillaci, Neville, Furness, Haines (Green 46’), Graham, Ayane (Wynne 85’), Dean, Quinn (Davison 57’).
Unused subs: Leon, Filbey, Peplow, Morgan.
Referee: Jane Simms.