Honestly, there's a few ways to assess victory, so let me go over those:

1. Tactically: Did the military of one side inflict more damage on the other than they had inflicted on them, and did they manage to take ground?

2. Politically: Did the actions alter the political balance in a favorable sense for one side, or the other?

3. Goal-wise: Did the sides in the war accomplish what they set out to accomplish?

Tactically

In a tactical sense, there's a vague numbers game, and a "Who got where" game. The Egyptians managed to initially make great gains, and the Israelis were hard-pressed to survive. So hard-pressed, actually, that they ordered a nuclear alert and armed 13 atomic bombs to essentially threaten Nixon into airlifting supplies from the US to keep the Israelis afloat. This threat succeeded, despite Soviet nuclear deterrence (the Soviets had placed a nuclear sub near the area, and a very visible ship), and Nixon proceeded to airlift supplies that were (by most accounts) crucial to winning the war. Despite the initial Egyptian/Syrian advantage (the main belligerents), which was predicated on the tactical surprise enjoyed by attacking on Israel's holiest day of non-work (many Jews don't use electricity, and fast without food or drink the entire day, as a reference), the Egyptian gains were eventually reversed. The Israelis had no idea the war was coming, however, which is why the airlift was necessary: they had trouble mobilizing their troops to stop the enemy, and the fact that they held out as long as they did has been described by some as a miracle in and of itself. To explain that, let's look at the plans and how well they were executed:

Egyptian plan: Called Operation Badr, it was to cross the Suez, proceed about 10-15 km east, and dig in. The Syrians didn't appreciate this plan, thinking Egypt should push further, so Egypt tricked them by drawing up a false plan that showed them going 48 km east of the Canal, which they didn't end up going for. To reinforce this, Egypt had 19 infantry brigades, 8 mechanized brigades, 10 armored brigades, 3 airborne brigades, an amphibious brigade, and an R-17E SSM brigade. This comes out to roughly 1,700 tanks, 2,000 armored vehicles, 2,500 artillery pieces, 1,500 anti-tank guns, 700 anti-tank guided weapons, several thousand RPG-7 anti-tank projectiles, and more than a thousand RPG-43 anti-tank grenades. Their air force had 400 fighter-bombers, 70 transport aircraft, 140 helicopters, 150 SAM battalions and 2,500 anti-aircraft guns. Their navy had 12 submarines, 5 destroyers, 3 frigates, 12 submarine chasers, 17 CSA and Komar class missile patrol boats, 30 Shershaen and P-6 motor torpedo boats, 14 minesweepers, and 14 landing craft.

This force was huge, and the anti-tank capability in particular would be a huge problem for Israel. They also had reinforcements from other nations (in smaller amounts), like extra MiG squadrons, armored brigades, and a few infantry brigades too. The Egyptians crushed the Bar-Lev line, which had some 505 Israeli troops (since the tank-crews who had been hit managed to join the fight) manning it. Israeli forces numbering 290 tanks in the Sinai when war began lost 153 almost immediately, and 35 Israeli planes were shot down by Egypt/Syria in the first 24 hours.

The problem for Egypt was that they, under pressure from the Syrians (who the Israelis were having far more luck counterattacking, unlike the disastrous first attempt for Israel at countering Egypt before concentrating forces properly) proceeded to try and take a further offensive. Once more a miscalculation on the Egyptian side helped the Israeli cause. The Egyptians moved forward, though they had no air-defence and the Israeli Air Force was still far superior. The results of this are clear, and written by Ariel Sharon (future prime minister of Israel, commanding a division at the time): "On Sunday October 14 at 06:20 massed Egyptian tank forces moved towards our lines. By early afternoon, 100–120 tanks of the Egyptian 21st armoured division were either flaming like torches or lying dead on the sand"

By midday throughout all the offensives, 250 tanks had been lost, more than double what they'd lost in the whole day prior. The Israelis proceeded to counterattack across the Suez, and by the end of the war (under pressure of threats from the Soviets and pressure from the Americans to make peace), were only 101 km from Cairo, had the enemy's forces more or less encircled or cut off from each other, and were in position to crush most of the Egyptian military. Just so you understand where the militaries stood, the border pre-war had been the Suez Canal, which was roughly 150 km from Cairo. Israel had now advanced 50 km into Egyptian territory, taken the military advantage, and threatened inflicting grave losses on Egyptian forces (the encircled numbered 35,000-45,000, and 250 tanks) with minimal losses of their own.

Syrian plan: Called Mashrua 110 (Operation 110), they planned to occupy the Golan Heights, establish bridgeheads west of the Jordan river, and move towards Nazareth in the Galilee. Syrian troops were also to take Israel's Hermon foothold, which was 2,100m above sea level and perfect therefore for observation of Syrian forces by the Israelis. The Syrians had 3 infantry divisions (45,000 men deployed). Each division had 2 infantry brigades, a tank brigade, and an armored brigade. Another division was in the north, ready to jump-off, with another division in the central sector, and another division along a line stretching from Rafid to the Yarmouk. 540 tanks were on the frontline, with 460 in reserve just behind the first line. The Republican Guard was a brigade meant to protect Damascus and was deployed in that mission, two armored brigades were ready to attack, and another 200 tanks were "static" and spread throughout the lines (meaning 1,500 tanks total were deployed, all these divisions and lines added up). There were also 942 artillery pieces ready, and again, reinforcements including tanks, MiGs, and more from other countries. They had 36 ground-to-air missiles ready, which provided a very effective air-defense system that could detect anything not flying below 500 ft (leaving the Israelis unable to fly over their land too, because it stretched 8 km into Israeli territory). This would inflict heavy losses on the IAF.

The IAF managed to strike at Syrian cities early in the war to demoralize them, mainly because the Syrians had momentarily run out of ground-to-air missiles. The Israelis then managed to turn back the Syrians, pushing back the 40th Jordanian tank brigade and then moving on the Iraqis, then proceeded to take back the Golan Heights. Defence Minister Moshe Dayan noted that now it was the Syrians on the defensive on their soil, east of the lines they had broken 6 days earlier, and this led to Egypt's aforementioned offensive that failed.

Israel ended up a mere 45 km from Damascus, whereas the Golan Heights are roughly 80 km from Damascus to begin with.

Israeli plan: The plan to counter Egypt was called "Shovach Yonim" (Operation Dovecote), and was fairly unimaginative. It was essentially to divide up the area into three (north, middle, south), and repel any attacks. There would be three lines, the Bar Lev line (180m wide, rising 20m on a sloping dune usually, and on the canal's line), a second line (5-8 km back into the Sinai, three battalions with 40 tanks per, and a battalion in each north, middle, south sector), and a third line (19-32 km back, based on reserves and 3 armored brigades with 120 tanks to a brigade). The plan was to, if Egypt attacked, move the second line up to the first to fire, and move the third line up to the second line to hold there with two formidable lines of defense. However, Israel was caught by surprise, so reserves and these lines and plans really couldn't come to fruition. Instead, the entire line was held by 10 infantry platoons, 52 cannons, 290 tanks, 2 ground-to-air Hawk missiles, and 6 anti-aircraft batteries. 10 infantry platoons was 450 troops, spread across 16 strongholds, and 4 observation points. Ariel Sharon, OC Southern Command at the time, didn't believe in the Bar-Lev line and so chose not to reinforce it, believing Israel should be more fluid and flexible in the event of an attack than rigid along those lines (he was proved right). Because of this, though, the Egyptians could essentially move between the strongholds, since 14-16 of the strongholds were shut down by Sharon. They advanced pretty much unopposed, to areas where they had geographical advantages in defending and could dig in/fortify.

The Israeli forces facing the Syrians were 10 infantry platoons, 178 tanks, and 44 artillery pieces. This was bigger than the standard force on the heights, because of a skirmish that had taken place days before between Syrian MiGs and Israeli Phantoms/Mirages (Israel went off to do photo-recon on the Syrians, the Syrians responded, fight ensued, 8 MiGs were shot down and 1 Mirage was shot down). Then the Israelis went to try to get their pilot, and ended up shooting down 4 more MiGs in a subsequent dogfight. Their plan, called Operation Chalk, was quite literally "Destroy anything that tries to take the Golan Heights". The Golan Heights were crucial for Israel, and they needed them to succeed (at least, in their eyes) because of their geographic significance for strategy.

All told, the Egyptians/Syrians lost some 15,600 men, with 35,000 wounded and 8,700 captured, to Israel's losses of 2,687, with 7,251 wounded and 314 taken prisoner. The Arabs lost 440 planes to Israel's 102. They lost 2,250 tanks to Israel's 400 destroyed, and 600 needing repairs. They lost 770 cannons to Israel's 25. 12 missile boats were sunk, the Israelis lost none. The Israelis took ground, the Arabs lost it.

Tactically, militarily, the Israelis won a resounding victory. Now, let's look at politically.