Tony Andreu curls a superb shot into the far corner from the edge of the penalty area as United look to have won the Irn Bru Cup semi-final before the half-time whistle.
Partick Thistle manager Alan Archibald on BBC Radio Scotalnd: "It was a massive game and a massive three points.
"There wasn't a lot of good football on show - I think because of the size of the game and what was at stake.
"It is a boost (being in the top six). We don't harp on about it too much - I think the last time it lasted three days.
"I said to the lads we've nine huge games before the split, let's win the home ones and see where it takes us, we've done that today and hopefully we can build on it."
There aren't many signs at Palmerston Park that Queens have the ability to force themselves back into the Irn Bru Cup semi-final.
United are comfortable as they look to protect the early lead thanks to goals from Charlie Telfer and Scott Fraser.
Scotland's pitches are taking a battering.
Celtic manager Brendan Rodgers has described the Premiership leaders' pitch as "old" and hinted that it will be dug up at the end of the season.
Meanwhile, Hearts' Tynecastle surface certainly hasn't been helping new head coach Ian Cathro's aim of playing passing football and there was a pow-wow on the pitch after today's draw with Inverness Caledonian Thistle.
Runaway League Two leaders Forfar Athletic suffered a blow with a 3-2 defeat away to Berwick Rangers.
Arbroath could only move to within eight points of their Angus rivals after being held at home to Stirling Albion.
Third-top Elgin City hammered Clyde 4-1 as Barry Ferguson's Cumbernauld side continued their slide down the table and look out of the promotion play-off contention.
Annan Athletic stay fourth with a 1-0 win over Cowdenbeath, who are four points adrift at the bottom.
Montrose are three points ahead of Berwick in fifth after a 1-1 draw away to Edinburgh City.
Hibs manager Neil Lennon has lambasted his team, despite a draw away to Raith Rovers extending their lead at the top of the Championship.
"We were a disgrace and totally disrespected the opponent today," said Lennon, whose side have a Scottish Cup replay at home to city rivals Hearts on Wednesday.
"I am raging. I didn't even celebrate the goal even though it was a great goal.
"They will all want to play on Wednesday, but they didn't want to play today."
United appear to be on their way to the final of the Irn Bru Cup after only 11 minutes as Scott Fraser fires through a sea of defenders from 12 yards.
In League One, Livingston have stretched their lead at the top to 12 points with a thumping 4-0 win away to second-top Airdrieonians in the battle between the two full-time sides.
Alloa Athletic leapfrog the Diamonds into second with their 2-0 win away to Queen's Park.
East Fife take advantage to move into fourth with a 3-2 win over Brechin City, who are now sixth.
It is as it was at the bottom, where Stenhousemuir and Stranraer remain tied on points after Stenny's 1-1 draw with Albion Rovers and Stranraer's 3-3 draw at home to Peterhead.
The Terrors make an early breakthrough in Dumfries as the Queens defence fail to clear a cross from the right and Charlie Telfer's scuffed shot slides past the goalkeeper from 15 yards.
St Johnstone manager Tommy Wright on BBC Radio Scotland: "We deserved to win it; I thought we were excellent.
"We started really well, dominated the ball, hit the post, maybe got a bit of fortune with the own goal.
"First half, they had a couple of long-range shots and we didn’t really feel under any pressure.
"Second half, we knew they’d put the ball in the box, but we defended well and on the counterattack we had numerous opportunities, the final ball just let us down.
"Scott Fox has made an incredible save from MacLean, we’ve hit the bar and the post off corners, we probably should’ve had a penalty, so to lose a goal the way we did – alright, it was good play getting to the situation where they got a shot off – was disappointing for the players, but they showed great character.
"We can’t rest, we can’t think we’re in the top six, we’ve got to go and do it week in, week out."
Our reporter Sandra Brown says there is a poor turnout at Palmerston Park for the Irn-Bru Cup semi-final, which has just kicked off in Dumfries.
Maybe the fans are staying home to watch it live on TV, with the game being broadcast on BBC Alba.
Queen of the South: Robinson, Hamill, Marshall, Dowie, Brownlie, Higgins, Lyle, Dobbie, Jacobs, Thoms, Murray.
Substitutes: Hooper, Atkinson, Dykes, Bell, Fergusson, Carmichael.
Dundee United: Zwick, Dillon, Dixon, Murray, Flood, Van der Struijk, Fraser, Telfer, Edjenguele, Nicholls, Andreu.
Substitutes: Long, Durnan, Allardice, Mikkelson, Coote, Smith, Murdoch.
Celtic manager Brendan Rodgers on BBC Radio Scotland: "I thought the players were typically very good on a difficult surface.
"Motherwell were never going to come here and open the game up.
"The first 20 minutes was a bit slow, but we still got our two goals.
"James Forrest was the man of the match today. He was outstanding.
"Overall, another clean sheet and we were very solid."
Motherwell manager Mark McGhee on BBC Radio Scotland: "Given Celtic's recent form, with a decent performance and 2-0 defeat, I was quite satisfied.
"The Aberdeen game was a total aberration and we needed to come today and stand up to it and we did."
Relegation is closing in on St Mirren despite the Paisley side not playing today.
Raith Rovers and Ayr United both picked up a point to move further clear of the bottom side.
Ayr scored twice in stoppage time to take a point against Dumbarton despite being down to 10 men for most of the game.
Rovers got theirs in their first game under John Hughes with a draw at home to leaders Hibs.
The Edinburgh side move seven points ahead of Dundee United.
Morton and Falkirk drew a thrilling game in Greenock and are locked on 41 points, with the Bairns ahead of goal difference.
Dumbarton 2-2 Ayr United
Greenock Morton 2-2 Falkirk
Raith Rovers 1-1 Hibernian
Airdrieonians 0-4 Livingston
Albion Rovers 1-1 Stenhousemuir
East Fife 3-2 Brechin City
Queen's Park 0-2 Alloa Athletic
Stranraer 3-3 Peterhead
Arbroath 1-1 Stirling Albion
Berwick Rangers 3-2 Forfar Athletic
Cowdenbeath 0-1 Annan Athletic
Edinburgh City 1-1 Montrose
Elgin City 4-1 Clyde
As Celtic stretch their lead at the top of the Premiership to 27 points, Hearts' missed penalty means they squander a chance to move to within three points of third-top Rangers.
Instead, Hearts are left looking behind them as St Johnstone's win away to Ross County moves the Perth side to within a point of the Edinburgh outfit.
Partick Thistle's 2-0 win over Hamilton takes the Glasgow side into sixth place ahead of Kilmarnock and Dundee, who play tomorrow, County and Motherwell.
Caley Thistle's point takes Richie Foran's side to within two points of second-bottom Accies.
Final score is on your telly...
A 20th straight league win for Celtic and they are again 27 points clear at the top of the Premiership.
Moussa Dembele and James Forrest were on target for the hosts.Copyright: SNS