Whoever catches one rakʿah of the Prayer just before the end of its time has caught the Prayer.

It is not permissible to delay the Prayers of ʿAsr and Fajr until it is almost too late to pray them―but if for some reason one finds himself in that situation (maybe he forgot to pray or fell asleep), then they can still pray them right at the end of the Prayer time to avoid missing them altogether. Abu Hurayrah (radiyallāhu ʿanhu) narrated that Allah’s Messenger (salallāhu ʿalaihi wasallam) said, “Whoever catches a Rakʿah of the Fajr prayer before the sun rises has caught the Fajr Prayer―and whoever catches a rakʿah of the ʿAsr prayer before the sun sets has caught the ʿAsr prayer.”

عَنْ أَبِي هُرَيْرَةَ، أَنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم قَالَ ‏ مَنْ أَدْرَكَ مِنَ الصُّبْحِ رَكْعَةً قَبْلَ أَنْ تَطْلُعَ الشَّمْسُ فَقَدْ أَدْرَكَ الصُّبْحَ، وَمَنْ أَدْرَكَ رَكْعَةً مِنَ الْعَصْرِ قَبْلَ أَنْ تَغْرُبَ الشَّمْسُ فَقَدْ أَدْرَكَ الْعَصْرَ ‏

Al-Bukhārī no. 579, Muslim no. 608, Abu Dāwūd no. 412, Al-Nasā’ī no. 514, al-Tirmidhī no. 186, Ibn Mājah no. 699, Ahmad 2/474.

Abu Khadeejah ʿAbdul-Wāhid.