However, Murray halved the deficit in the 67th minute before converting Brighton's second penalty of the match in the 84th minute to ensure that the points would be shared.
Both teams had their moments in a bright opening 16 minutes, but Fulham had the best chance during that period, with Mitrovic heading a Schurrle cross over the Brighton goal.
The hosts had a golden chance to take the lead in the 24th minute of the match, however, when Luciano Vietto brought Murray to the deck inside the Fulham box.
Gross scored against Manchester United from the spot last month, but the midfielder saw his effort saved by Fulham goalkeeper Marcus Bettinelli on this occasion.
It was Fulham that made the breakthrough just before the interval, with Jean Michael Seri's excellent chip finding Schurrle, and the summer arrival poked the ball into the back of the net.
Fulham then doubled their advantage in the 62nd minute when Mitrovic made the most of a mistake from Lewis Dunk to finish past Ryan for his fourth Premier League goal of the campaign.
Brighton had one back five minutes later though, with Murray collecting a pass from Anthony Knockaert before finishing into the back of the net to set up a grandstand finish at the Amex Stadium.
Bettinelli had to keep out a header from Jurgen Locadia in the period that followed as Brighton looked for a leveller, and it was to come from their second spot kick of the match late on.
Indeed, Mitrovic was penalised for a handball inside his own box, and Murray stepped forward to send the ball into the back of the net.
Both teams had chances to win it late on, but Murray was denied a hat trick by a smart save from Bettinelli, while Ryan Sessegnon could not find a route past Ryan as the pair played out a 2-2 draw.
BRIGHTON (4-4-1-1): Ryan; Montoya, Duffy, Dunk, Bong; March, Propper (Locadia 74'), Stephens, Knockaert (Jahanbaksh 74'); Gross (Bissouma 60'); Murray
FULHAM (4-3-3): Bettinelli; Fosu-Mensah, Odoi, Maison, Le Marchand; Zambo (Johansen 74'), McDonald, Seri; Vietto (R Sessegnon 74'), Mitrovic, Schurrle (Chambers 87')