Hearts sealed third place in the Scottish Premier League with victory over Dundee United, who can still join their victors in the Europa League.
Lee Wallace rifled home to put Hearts in front after 11 minutes.
Bruno Aguiar added a second before half-time when he converted Calum Elliot's cross.
Elliot was fouled in the box by Garry Kenneth and Eggert Jonsson converted the resulting penalty with 11 minutes of the match remaining.
Aberdeen's defeat by Rangers earlier in the day had already guaranteed Hearts qualification to next season's Europa League and the win edged them five points clear of fourth-place United with one round of fixtures left.
However, the visitors are still on course to claim the remaining European place with a three-point advantage over the Dons, who have the same goal difference as United.
Bruno Aguiar drew a save from Dundee United goalkeeper Lukasz Zaluska before David Obua's effort was deflected wide during a period of early Hearts pressure.
And it was not long until the home side edged themselves in front as Wallace robbed United defender Sean Dillon and fired beyond Zaluska.
David Goodwillie attempted to draw his side level with an inventive flick but goalkeeper Marian Kello was equal to the United striker's attempt.
Dead ball specialist Bruno Aguiar was given the opportunity to stretch Hearts' lead with a free-kick but Zaluska tipped the Portuguese midfielder's effort over.
But Aguiar did find the net minutes later when he beat team-mate Michael Stewart to Calum Elliot's cross before hooking the ball home.
United manager Craig Levein made a substitution to try to change the course of the match as Paul Caddis replaced Dillon.
Goodwillie continued to cause Hearts problems and his shot was tipped over by Kello minutes before half-time.
Hearts looked to put the result beyond doubt when Wallace crossed towards Elliot but Kenneth headed clear under pressure.
Further substitutions followed for United with Danny Swanson and Andis Shala sent on in place of David Robertson and Francisco Sandaza.
Stewart tested Zaluska and Elliot blasted over either side of Prince Buaben's wayward header at the other end.
Elliot was wildly off target with a left-foot volley before winning the penalty which sealed Hearts' win.
Jonsson showed composure to send Zaluska the wrong way.
Substitute Christian Nade shot over with several team-mates awaiting a pass in the penalty area as Hearts pressed for a fourth.
And as the match wound down, the home fans gave a standing ovation to Omonia Nicosia-bound defender Christos Karipidis, whose final game at Tynecastle was brought to a close when he was replaced by Jason Thomson.
Hearts manager Csaba Laszlo: "It is good feeling but we don't forget our opponents. We had very fair, very strong opponents who always play a fair game.
"We had the luck, especially in the first half to shoot two goals.
"Altogether, this season was great for the fans, for the players and also for me.
"We must look forward. We must look at which players' contracts are going to run out, handle it very quickly and keep the team together.
"If we don't keep the team together, this season was for nothing. We must bring in new blood that will bring quality to Tynecastle."
Dundee United manager Craig Levein: "They did deserve to win, I cannot argue with that. We didn't do enough today. In all areas of the pitch, we weren't at our best. I'm bitterly disappointed.
"The biggest disappointment, of course, is the magnitude of the game and what it meant today. We just didn't have enough about us to compete at the level we would've needed to win the game.
"It's a combination of things. We didn't defend properly. We had opportunities to track runners at one-twos and we didn't do it properly.
"At the second goal we had the chance to attack the ball and we didn't do that and let somebody get across in front of us. That affects the confidence.
"We gave away the ball on numerous occasions under no pressure. And when we got into good positions, we hit the first man with our crosses.
"You put all those things together and it's pretty evident that you're not going to win a football match."
Hearts: Kello, Neilson, Jonsson, Karipidis (Thomson 90), Lee Wallace, Obua, Aguiar (Templeton 86), Michael Stewart (Nade 82), Palazuelos, Elliot, Mrowiec. Subs Not Used: Balogh, Tullberg, Johnny Stewart, Novikovas.
Goals: Lee Wallace 11, Aguiar 26, Jonsson 79 pen.
Dundee Utd: Zaluska, Dillon (Caddis 31), Wilkie, Kenneth, Dixon, David Robertson (Swanson 62), Buaben, Gomis, Conway, Sandaza (Shala 61), Goodwillie. Subs Not Used: McGovern, Scott Robertson, Grainger, Dods.
Booked: Sandaza, Kenneth, Swanson.
Att: 15,664
Ref: B Winter
BBC Sport Player Rater man of the match: Hearts' Bruno Aguiar 8.80 (on 90 minutes).
Please note that you can still give the players marks out of 10 on BBC Sport's Player Rater after the match has finished.
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