Madrid reeled league leaders Barcelona’s advantage back to four points and recorded only their third victory in the last nine games across all competitions.
After a home defeat by Manchester City in the Champions League on Wednesday, Spanish media reported anything but a victory would cost Alonso his position.
After Mbappe’s superb opener, Carlos Vicente pulled Alaves level in the second half, but Rodrygo secured the visitors a much-needed victory at Mendizorroza stadium.
Alonso was able to bring his key player, Mbappe, back into the side after he could only watch the defeat by City from the bench because of knee discomfort.
The coach also handed a debut to Victor Valdepenas at left-back, with both Alvaro Carreras and Fran Garcia suspended, and Ferland Mendy one of several players out injured.
Mbappe appeared to be feeling his knee and also hobbling in the first few minutes but, despite that, was the game’s most influential player.
The forward had a shot deflected wide and then fired narrowly over as Alaves sat deep and tried to keep the 15-time European champions at bay.
By the time Mbappe opened the scoring in the 25th minute his discomfort seemed to have cleared up.
Released by Jude Bellingham, Mbappe drove towards goal at full tilt and whipped a shot into the top right corner for his 17th league goal of the campaign.
England international Bellingham then blasted home from close range but his strike was ruled out for handball, as the ball had struck his arm.
Needing to fight back, Alaves moved on to the front foot and took control of the game before the break, almost pulling level.
Madrid goalkeeper Thibaut Courtois made a fine save with his head, even if he knew little about it, to deny Pablo Ibanez from close range. (AFP)