The French superstar was thwarted by Mallorca goalkeeper Dominik Greif and after Rodrygo Goes sent Madrid ahead early on the hosts levelled in the second half through Vedat Muriqi's towering header.
Madrid's star-studded side struggled to break down Mallorca, who came closer to winning the game in the final stages at Son Moix stadium.
Los Blancos defender Ferland Mendy was sent off in stoppage time for a high challenge on Muriqi.
"We went ahead, we had chances to score the second, and then in the second half we lacked balance," Real Madrid coach Carlo Ancelotti told reporters.
"We conceded counter-attacks, it was not a good game, it was quite a clear game to see that we need to defend better and have better balance on the pitch."
Ancelotti selected the same starting line-up which defeated Atalanta to win the UEFA Super Cup on Wednesday, with Jude Bellingham in midfield behind the Mbappe, Vinicius Junior and Rodrygo attacking line.
Mbappe scored in that victory but the former Paris Saint-Germain forward was unable to celebrate his first match in his new homeland with another goal.
Mallorca coach Jagoba Arrasate was facing his first match in charge of the Copa del Rey runners-up and could not have asked for a tougher challenge on paper than the reigning champions, bolstered by Mbappe's arrival.
However, his team rose to the challenge and earned a creditable point.
Madrid, who lost just once last season as they marched convincingly to the title, started quickly and took the lead after a slick combination between their Brazilian forwards.
Vinicius rolled the ball backwards for Rodrygo to collect in the area and the winger bent a shot into the far corner, beyond the reach of Greif.
The goalkeeper tipped away another Rodrygo effort and Mbappe sent an effort wide as Madrid stayed on top but could not expand upon their advantage.
Arrasate urged his players to stay in the game until half-time and they duly obliged, shutting down Los Blancos' array of stars.
'So beautiful'
Muriqi came close before the break for the hosts and tough-tackling defender Pablo Maffeo was booked for an ugly challenge on Vinicius.
Mallorca levelled early in the second half with burly Kosovo target man Muriqi powering a header home from Dani Rodriguez's corner.
Greif saved low from Mbappe at his near post and Vinicius lashed over as Ancelotti's side sought to regain the lead.
Ancelotti sent on veteran Luka Modric, who turns 39 in September, for defensive midfielder Aurelien Tchouameni, in search of more fluidity in attack against Mallorca's bunkered-in defence.
Mbappe came close again but Greif denied him once more and Johan Mojica hacked the rebound to safety before Rodrygo could reach it.
Mallorca stabilised and held their own in the final stages to leave Mbappe disappointed on his league debut.
Madrid's frustration showed in Mendy's wild hack at Muriqi which earned him a straight red card.
"I'm so tired but it was worth it, it was so important to get a point against a team like Madrid, maybe the best in the world," said Muriqi.
"I have to be honest, scoring against Madrid is so beautiful."
