Fulham were frustrated in Marco Silva's first game in charge as Marc Bola cancelled out Harry Wilson's debut goal to earn Middlesbrough a point.
Wilson, a £12m signing from Liverpool, put the hosts ahead with a left-footed strike from just outside the box.
Fulham dominated possession but were guilty of missing several chances to double their lead in the second half.
They were made to pay when Bola fired home with 13 minutes remaining from Boro's only shot on target.
The result means none of the three teams relegated from the Premier League last season won their opening fixture back in the second tier after West Bromwich Albion's draw at Bournemouth on Friday and Sheffield United's defeat by Birmingham City on Saturday.