Heart of Midlothian 0 Falkirk 1
Hearts claimed Lovell was in an offside position when he headed home after 59 minutes.
Marius Zaliukas for Hearts and Scott Arfield for Falkirk were both sent off in a frantic final few minutes of a match sprinkled with controversy.
Dani Mallo was called into action after only three minutes when as back heel from Andrew Driver gave Lee Wallace a chance but his effort was pushed away by the goalkeeper.
Gary Glen should have given Hearts the lead midway through the first half when Arfield failed to deal with a cross.
The ball broke to Andrew Driver who cut the ball back for Glen, but he shot wide from only eight yards.
Janos Balogh did well to leave his line quickly to thwart Lovell after Arfield had split the Hearts defence.
There was a let-off for Hearts eight minutes from half-time when Balogh came off his line and did not get to a Lee Bullen free-kick but Michael Higdon headed wide.
Malo produced a brilliant save four minutes from the break when a cutback from Bruno Aguiar gave Glen another chance but the young striker saw his effort tipped over the top.
Higdon had the first chance of the second half for Falkirk when he held off Hristos Karipidis but he then sent his 18-yard shot high over the top.
Falkirk took the lead after 59 minutes with a disputed goal. Lovell looked to be in an offside position when he headed on Jackie McNamara's free-kick but referee Dougie MacDonald awarded the strike.
MacDonald then enraged Hearts fans when he awarded a penalty after Dean Holden tackled Driver in the area only to change his mind and award a corner after consulting his linesman.
Bullen had a chance to make it 2-0 with a back-post header which Balogh misjudged but the ball slipped past the post.
There was a let-off for Falkirk when Neil McCann stabbed the ball back and Mallo had to touch the ball onto his crossbar.
Zaliukas was then rightly sent off as he wrestled Carl Finnigan to the ground as he tried to get on the end of a Steven Pressley clearance with only Balogh to beat.
Arfield followed him as he picked up his second booking in injury-time.