Shalom told the annual UN General Assembly debate that a seat on the most powerful UN decision-making body would help Israel "take its rightful place, as a country with full and equal rights in this institution".
Israel's relationship with the United Nations, long strained, has improved in the last year with a seminar to address anti-Semitism, and a special General Assembly session to commemorate the 67th anniversary of the liberation of Nazi concentration camps.
That new warmth culminated in June when Israel was elected one of 21 vice-presidents for the 60th General Assembly session.
While seemingly a small step, the move had symbolic resonance because Israel had long been denied a presence on key UN bodies.
"Israel's relations with the UN are better today than they have ever been," Shalom said.
Shalom may also believe he can capitalise on Israel's recent withdrawal from the Gaza Strip.
In his speech, he said the Middle East was seeing "optimistic times", a notion that was encouraged by several meetings between Israeli and Arab leaders in New York over the last week.
The earliest Israel can vie for one of 10 non-permanent, rotating spots on the 15-member council is 2017, Israeli mission spokeswoman Anat Friedman said.
The non-permanent seats are divided among regional groups and usually determined well in advance.
Russia, China, the US, France and Britain occupy the other five seats, which are permanent and hold veto power.
Despite an improved atmosphere, it is still likely that many Arab and Islamic states would still bridle at Israel getting a Security Council seat and probably try to block the move.
"It is not going to be easy, but we hope it will be reachable," Friedman said.